{"type":"doc","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"{\"type\":\"doc\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q1. Give a geometric description of the set of points in space whose coordinates satisfy y = 0, z = 0.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Geometry in 3D Space\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your understanding of how equations define geometric objects in three-dimensional space.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Coordinate planes: Planes defined by setting one or more coordinates to a constant value.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Intersection: The set of points that satisfy multiple equations simultaneously.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Consider the equation \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 0\"}},{\"type\":\"text\",\"text\":\". This describes all points where the y-coordinate is zero, which is the xz-plane.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Consider the equation \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 0\"}},{\"type\":\"text\",\"text\":\". This describes all points where the z-coordinate is zero, which is the xy-plane.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"To find the set of points that satisfy both \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 0\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 0\"}},{\"type\":\"text\",\"text\":\", look for the intersection of these two planes.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Think about what points have both \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 0\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 0\"}},{\"type\":\"text\",\"text\":\" simultaneously. What does this look like geometrically?\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The set of points is the x-axis in three-dimensional space, where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 0\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 0\"}},{\"type\":\"text\",\"text\":\" for all \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Geometrically, this is a straight line along the x-axis.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q2. Describe the circle of radius 1 centered at (0, 8, 0) in:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"a. The xy-plane\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"b. The yz-plane\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"c. The plane y = 8\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Equations of Circles in 3D Space\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to write equations for circles in various planes in three-dimensional space.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Circle equation in 2D: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x - h)^2 + (y - k)^2 = r^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"In 3D, you must specify the plane in which the circle lies.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each part, identify which coordinates are fixed by the plane.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the circle equation using the center \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(0, 8, 0)\"}},{\"type\":\"text\",\"text\":\" and radius $1"},{"type":"inlineMath","attrs":{"latex":".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For the xy-plane, \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 0\"}},{\"type\":\"text\",\"text\":\"; for the yz-plane, \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 0\"}},{\"type\":\"text\",\"text\":\"; for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 8\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\" is fixed.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the equations for each case, but leave the final form for you to complete.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"a. In the xy-plane (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 0\"}},{\"type\":\"text\",\"text\":\"): \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x - 0)^2 + (y - 8)^2 = 1\"}},{\"type\":\"text\",\"text\":\", "}},{"type":"text","text":"z = 0"},{"type":"inlineMath","attrs":{"latex":"\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"b. In the yz-plane (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 0\"}},{\"type\":\"text\",\"text\":\"): \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(y - 8)^2 + (z - 0)^2 = 1\"}},{\"type\":\"text\",\"text\":\", "}},{"type":"text","text":"x = 0"},{"type":"inlineMath","attrs":{"latex":"\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"c. In the plane \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 8\"}},{\"type\":\"text\",\"text\":\": \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x - 0)^2 + (z - 0)^2 = 1\"}},{\"type\":\"text\",\"text\":\", "}},{"type":"text","text":"y = 8"},{"type":"inlineMath","attrs":{"latex":"\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Each equation describes a circle of radius 1 centered at (0, 8, 0) in the specified plane.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q3. Let \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} = \\\\langle 4, -1 \\\\rangle\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\". Find the component form and magnitude (length) of the vector \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2\\\\mathbf{u}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Vector Operations\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to scale a vector and compute its magnitude.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Component form: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} = \\\\langle u_1, u_2 \\\\rangle\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Scaling: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"k\\\\mathbf{u} = \\\\langle k u_1, k u_2 \\\\rangle\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Magnitude: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{u}| = \\\\sqrt{u_1^2 + u_2^2}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply each component of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u}\"}},{\"type\":\"text\",\"text\":\" by 2 to get \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2\\\\mathbf{u}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the component form of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2\\\\mathbf{u}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Use the magnitude formula to set up the calculation for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|2\\\\mathbf{u}|\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug in the values, but stop before computing the final square root.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Component form: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2\\\\mathbf{u} = \\\\langle 8, -2 \\\\rangle\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Magnitude: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|2\\\\mathbf{u}| = \\\\sqrt{8^2 + (-2)^2} = \\\\sqrt{64 + 4} = \\\\sqrt{68} \\\\approx 8.25\"}}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q4. Express the vector \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\overrightarrow{AB}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" in the form \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v = v_1 \\\\mathbf{i} + v_2 \\\\mathbf{j} + v_3 \\\\mathbf{k}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" if \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"A\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" is the point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-3, -7, 4)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"B\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" is the point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(10, 7, 4)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Vectors Between Points in 3D\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to find the vector from one point to another in three-dimensional space.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Vector from \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"A\"}},{\"type\":\"text\",\"text\":\" to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"B\"}},{\"type\":\"text\",\"text\":\": \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\overrightarrow{AB} = \\\\langle x_B - x_A, y_B - y_A, z_B - z_A \\\\rangle\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Component form: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v = v_1 \\\\mathbf{i} + v_2 \\\\mathbf{j} + v_3 \\\\mathbf{k}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Subtract the coordinates of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"A\"}},{\"type\":\"text\",\"text\":\" from \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"B\"}},{\"type\":\"text\",\"text\":\" for each component.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the resulting vector in component form.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Express the vector using \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{i}\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{j}\"}},{\"type\":\"text\",\"text\":\", and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{k}\"}},{\"type\":\"text\",\"text\":\" notation.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check if any components are zero and simplify accordingly.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\overrightarrow{AB} = \\\\langle 13, 14, 0 \\\\rangle = 13\\\\mathbf{i} + 14\\\\mathbf{j}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z\"}},{\"type\":\"text\",\"text\":\"-component is zero since both points have \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 4\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q5. For the vectors \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} = 4\\\\mathbf{i} - 3\\\\mathbf{k}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} = \\\\mathbf{i} + \\\\mathbf{j} + \\\\mathbf{k}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", find:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"a. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} \\\\cdot \\\\mathbf{u}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"b. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{v}|\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"c. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{u}|\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"d. The cosine of the angle between \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"e. The scalar component of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u}\"}},{\"type\":\"text\",\"text\":\" in the direction of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"f. The vector projection \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathrm{proj}_{\\\\mathbf{v}} \\\\mathbf{u}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Dot Product, Magnitude, Projections\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your understanding of vector operations, including dot product, magnitude, angle, and projections.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Dot product: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} \\\\cdot \\\\mathbf{u} = v_1 u_1 + v_2 u_2 + v_3 u_3\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Magnitude: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{v}| = \\\\sqrt{v_1^2 + v_2^2 + v_3^2}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Cosine of angle: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\theta = \\\\frac{\\\\mathbf{v} \\\\cdot \\\\mathbf{u}}{|\\\\mathbf{v}| |\\\\mathbf{u}|}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Scalar component: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathrm{comp}_{\\\\mathbf{v}} \\\\mathbf{u} = \\\\frac{\\\\mathbf{v} \\\\cdot \\\\mathbf{u}}{|\\\\mathbf{v}|}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Vector projection: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathrm{proj}_{\\\\mathbf{v}} \\\\mathbf{u} = \\\\frac{\\\\mathbf{v} \\\\cdot \\\\mathbf{u}}{|\\\\mathbf{v}|^2} \\\\mathbf{v}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write out the components for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate the dot product \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} \\\\cdot \\\\mathbf{u}\"}},{\"type\":\"text\",\"text\":\" using the formula.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the magnitude calculations for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{v}|\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{u}|\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the formula for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\theta\"}},{\"type\":\"text\",\"text\":\" using the values from previous steps.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the scalar component and vector projection formulas, plugging in the dot product and magnitude values.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Stop before plugging in the final numbers for each part.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"a. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} \\\\cdot \\\\mathbf{u} = 4 \\\\times 1 + 0 \\\\times 1 + (-3) \\\\times 1 = 4 + 0 - 3 = 1\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"b. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{v}| = \\\\sqrt{4^2 + 0^2 + (-3)^2} = \\\\sqrt{16 + 0 + 9} = \\\\sqrt{25} = 5\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"c. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{u}| = \\\\sqrt{1^2 + 1^2 + 1^2} = \\\\sqrt{3} \\\\approx 1.732\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"d. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\theta = \\\\frac{1}{5 \\\\times \\\\sqrt{3}} \\\\approx 0.115\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"e. Scalar component: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{1}{5} = 0.2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"f. Vector projection: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{1}{25} \\\\mathbf{v} = \\\\frac{1}{25}(4\\\\mathbf{i} - 3\\\\mathbf{k}) = 0.16\\\\mathbf{i} - 0.12\\\\mathbf{k}\"}}]}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q6. Find the angle between the vectors in radians: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} = \\\\sqrt{3}\\\\mathbf{i} - 5\\\\mathbf{j}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} = \\\\sqrt{3}\\\\mathbf{i} + \\\\mathbf{j} - 2\\\\mathbf{k}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Angle Between Vectors\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to use the dot product to find the angle between two vectors.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Dot product: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} \\\\cdot \\\\mathbf{v}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Magnitude: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{u}|\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{v}|\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Angle formula: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\theta = \\\\frac{\\\\mathbf{u} \\\\cdot \\\\mathbf{v}}{|\\\\mathbf{u}| |\\\\mathbf{v}|}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"To find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\theta\"}},{\"type\":\"text\",\"text\":\", use \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\theta = \\\\arccos(\\\\cos \\\\theta)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write out the components for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u}\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate the dot product \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} \\\\cdot \\\\mathbf{v}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the magnitude calculations for both vectors.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug the dot product and magnitudes into the cosine formula.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the arccos calculation to find the angle in radians, but stop before computing the final value.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Dot product: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} \\\\cdot \\\\mathbf{v} = (\\\\sqrt{3})(\\\\sqrt{3}) + (-5)(1) + (0)(-2) = 3 - 5 + 0 = -2\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{u}| = \\\\sqrt{(\\\\sqrt{3})^2 + (-5)^2} = \\\\sqrt{3 + 25} = \\\\sqrt{28}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{v}| = \\\\sqrt{(\\\\sqrt{3})^2 + 1^2 + (-2)^2} = \\\\sqrt{3 + 1 + 4} = \\\\sqrt{8}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\theta = \\\\frac{-2}{\\\\sqrt{28} \\\\cdot \\\\sqrt{8}}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\theta = \\\\arccos\\\\left(\\\\frac{-2}{\\\\sqrt{28} \\\\cdot \\\\sqrt{8}}\\\\right) \\\\approx 1.68\"}},{\"type\":\"text\",\"text\":\" radians\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q7. Find the length and direction (when defined) of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} \\\\times \\\\mathbf{v}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} \\\\times \\\\mathbf{u}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} = 5\\\\mathbf{i} + 6\\\\mathbf{j} - 4\\\\mathbf{k}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} = 0\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Cross Product\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your understanding of the cross product, especially when one vector is the zero vector.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Cross product: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} \\\\times \\\\mathbf{v}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Magnitude: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{u} \\\\times \\\\mathbf{v}|\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Direction: The cross product is perpendicular to both vectors, but if one is zero, direction is undefined.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recall that the cross product of any vector with the zero vector is the zero vector.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write out the cross product formula and substitute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} = 0\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate the magnitude of the resulting vector.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Discuss whether direction is defined for the zero vector.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{u} \\\\times \\\\mathbf{v} = \\\\mathbf{0}\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v} \\\\times \\\\mathbf{u} = \\\\mathbf{0}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The length is $0"}},{"type":"text","text":", and direction is undefined for the zero vector.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q8. Find the area of the parallelogram whose vertices are given below: A(-1, 2), B(2, 0), C(9, 2), D(6, 4).\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Area of Parallelogram in 2D\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to use vectors to find the area of a parallelogram given its vertices.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Area formula: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\text{Area} = |\\\\mathbf{AB} \\\\times \\\\mathbf{AD}|\"}},{\"type\":\"text\",\"text\":\" (in 3D), or \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{AB}| \\\\cdot |\\\\mathbf{AD}| \\\\cdot \\\\sin \\\\theta\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"In 2D, area can be found using the determinant: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|a_1 b_2 - a_2 b_1|\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Choose one vertex (e.g., \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"A\"}},{\"type\":\"text\",\"text\":\") and find vectors \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{AB}\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{AD}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{AB} = B - A\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{AD} = D - A\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the determinant formula for area in 2D.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug in the vector components, but stop before computing the determinant.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{AB} = (2 - (-1), 0 - 2) = (3, -2)\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{AD} = (6 - (-1), 4 - 2) = (7, 2)\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Area \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"= |3 \\\\times 2 - (-2) \\\\times 7| = |6 + 14| = 20\"}}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q9. Find the parametric equations for the line shown below (x-axis).\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Parametric Equations of Lines\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to write parametric equations for a line, specifically the x-axis.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Parametric equation: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = x_0 + at\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = y_0 + bt\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = z_0 + ct\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For the x-axis, direction vector is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(1, 0, 0)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Choose a point on the x-axis, such as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(0, 0, 0)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the direction vector for the x-axis.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the parametric equations using the point and direction vector.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the equations for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\", and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z\"}},{\"type\":\"text\",\"text\":\" in terms of parameter \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Parametric equations: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = t\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 0\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = 0\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This describes all points on the x-axis.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q10. Find the equation for the plane through the point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"P_0 = (0, 3, -2)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and normal to the vector \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{n} = 2\\\\mathbf{i} - 8\\\\mathbf{j} - \\\\mathbf{k}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Equation of a Plane in 3D\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to write the equation of a plane given a point and a normal vector.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plane equation: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Normal vector: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(a, b, c)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify the normal vector components: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"a = 2\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"b = -8\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"c = -1\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug in the point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"P_0 = (0, 3, -2)\"}},{\"type\":\"text\",\"text\":\" into the plane equation.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the equation in terms of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\", and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify the equation, but stop before writing the final simplified form.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plane equation: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2(x - 0) - 8(y - 3) - 1(z + 2) = 0\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplified: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"2x - 8y - z = -22\"}}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q11. Find the distance from the point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"P(6, -8, -3)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" to the line \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 6 + 3t\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = -8 + 9t\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"z = -3\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Distance from Point to Line in 3D\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to use vector projection to find the shortest distance from a point to a line.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Distance formula: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"d = \\\\frac{|\\\\mathbf{a} \\\\times \\\\mathbf{b}|}{|\\\\mathbf{b}|}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{a}\"}},{\"type\":\"text\",\"text\":\": vector from point on line to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"P\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{b}\"}},{\"type\":\"text\",\"text\":\": direction vector of the line\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify a point on the line (e.g., \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = 0\"}},{\"type\":\"text\",\"text\":\" gives \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(6, -8, -3)\"}},{\"type\":\"text\",\"text\":\").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find the direction vector \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{b} = \\\\langle 3, 9, 0 \\\\rangle\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find the vector \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{a}\"}},{\"type\":\"text\",\"text\":\" from the point on the line to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"P\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the cross product \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{a} \\\\times \\\\mathbf{b}|\"}},{\"type\":\"text\",\"text\":\" and the magnitude \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"|\\\\mathbf{b}|\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Stop before computing the final value.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Since \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"P\"}},{\"type\":\"text\",\"text\":\" is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(6, -8, -3)\"}},{\"type\":\"text\",\"text\":\" and the line passes through "},{"type":"inlineMath","attrs":{"latex":"(6, -8, -3)"}},{"type":"text","text":", the distance is $0"},{"type":"inlineMath","attrs":{"latex":".\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The point lies on the line.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q12. Find the limit or indicate that it does not exist:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{t \\\\to \\\\pi} \\\\left[ \\\\left( \\\\frac{\\\\sin t}{3} \\\\right) \\\\mathbf{i} + \\\\left( \\\\cos \\\\frac{3}{4} t \\\\right) \\\\mathbf{j} + \\\\left( \\\\tan \\\\frac{2}{3} t \\\\right) \\\\mathbf{k} \\\\right]\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits of Vector Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate limits for vector-valued functions component-wise.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Limit of vector function: Take the limit of each component separately.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Trigonometric limits: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin t\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos t\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tan t\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write out each component and substitute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\pi\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin \\\\pi\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\frac{3}{4} \\\\pi\"}},{\"type\":\"text\",\"text\":\", and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tan \\\\frac{2}{3} \\\\pi\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check if any component is undefined at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\pi\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the limit expressions for each component, but stop before computing the final values.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin \\\\pi = 0\"}},{\"type\":\"text\",\"text\":\", so first component is $0"}},{"type":"text","text":".\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\frac{3}{4} \\\\pi = \\\\cos(135^\\\\circ) = -\\\\frac{\\\\sqrt{2}}{2}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tan \\\\frac{2}{3} \\\\pi = \\\\tan(120^\\\\circ) = -\\\\sqrt{3}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Limit: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\left[ 0 \\\\mathbf{i} - \\\\frac{\\\\sqrt{2}}{2} \\\\mathbf{j} - \\\\sqrt{3} \\\\mathbf{k} \\\\right]\"}}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q13. The path \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{r}(t) = (t - \\\\sin t)\\\\mathbf{i} + (1 - \\\\cos t)\\\\mathbf{j}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" describes motion on the cycloid \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = t - \\\\sin t\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 1 - \\\\cos t\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\". Find the particle’s velocity and acceleration vectors at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\pi\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and sketch them as vectors on the curve.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Velocity and Acceleration of Vector Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to differentiate vector functions and evaluate at a specific value.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Velocity: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}(t) = \\\\frac{d\\\\mathbf{r}}{dt}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Acceleration: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{a}(t) = \\\\frac{d^2\\\\mathbf{r}}{dt^2}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Differentiate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{r}(t)\"}},{\"type\":\"text\",\"text\":\" with respect to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t\"}},{\"type\":\"text\",\"text\":\" to find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}(t)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Differentiate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}(t)\"}},{\"type\":\"text\",\"text\":\" with respect to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t\"}},{\"type\":\"text\",\"text\":\" to find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{a}(t)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug in \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\pi\"}},{\"type\":\"text\",\"text\":\" into both \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}(t)\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{a}(t)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the expressions for each vector, but stop before computing the final values.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}(t) = (1 - \\\\cos t)\\\\mathbf{i} + (\\\\sin t)\\\\mathbf{j}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"At \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\pi\"}},{\"type\":\"text\",\"text\":\": \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{v}(\\\\pi) = (1 - (-1))\\\\mathbf{i} + (0)\\\\mathbf{j} = 2\\\\mathbf{i}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{a}(t) = (\\\\sin t)\\\\mathbf{i} + (\\\\cos t)\\\\mathbf{j}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"At \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\pi\"}},{\"type\":\"text\",\"text\":\": \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\mathbf{a}(\\\\pi) = (0)\\\\mathbf{i} + (-1)\\\\mathbf{j} = -\\\\mathbf{j}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Velocity vector points along the x-direction, acceleration points downward along the y-direction at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = \\\\pi\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":nul