Q: Given a vector $$\vec{A}$$ with a magnitude of $$12\,\mathrm{m} at an $$angle of $$37^\circ$$ above the x-axis, what is the x-component $$A_x of $$the vector?

$$A_x = 12 \times \cos(37^\circ)$$

Q: Given $$\vec{B}$$ has a magnitude of $$15\,\mathrm{m}$$ and points $$40^\circ$$ below the x-axis, what is the y-component $$B_y$$?

$$B_y = -15 \times \sin(40^\circ)$$

Q: A vector $$\vec{B}$$ has components $$B_x = 8\,\mathrm{m}$$ and $$B_y = -6\,\mathrm{m}. $$What is the direction angle $$\theta ($$measured from the positive x-axis)?

$$\theta = \arctan\left(\frac{-6}{8}\right)$$

Question 1

A hiker walks along a winding path from an initial position to a final position, as shown in the diagram. Which of the following statements best describes the difference between distance and displacement?

Accepted Answer

Distance is the total length of the path traveled, while displacement is the straight-line measurement from the initial to the final position.

Question 2

Given a vector $$\vec{A}$$ with a magnitude of $$12\,\mathrm{m} at an $$angle of $$37^\circ$$ above the x-axis, what is the x-component $$A_x of $$the vector?

Accepted Answer

$$A_x = 12 	imes \cos(37^\circ)$$

Question 3

A plane flies $$200\,\mathrm{mi} at$$ $$50^\circ$$ north of east. What is the northward (y) component of its displacement?

Accepted Answer

$$200 	imes \sin(50^\circ)$$

Question 4

Which quadrant does a vector point to if its x-component is negative and its y-component is positive?

Accepted Answer

Quadrant 2

Question 5

Given $$\vec{B}$$ has a magnitude of $$15\,\mathrm{m}$$ and points $$40^\circ$$ below the x-axis, what is the y-component $$B_y$$?

Accepted Answer

$$B_y = -15 	imes \sin(40^\circ)$$

Question 6

A vector $$\vec{C}$$ has a magnitude of $$6\,\mathrm{m}$$ and points $$60^\circ$$ below the negative x-axis. What are the signs of its x and y components?

Accepted Answer

x is negative, y is negative

Question 7

If $$\vec{A}$$ and $$\vec{B}$$ are two vectors, which equation correctly represents the vector subtraction $$\vec{A} - \vec{B}$$?

Accepted Answer

$$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$$

Question 8

A boat travels $$60\,\mathrm{km} at$$ $$70^\circ$$ north of east. What is the eastward (x) component of its displacement?

Accepted Answer

$$60 	imes \cos(70^\circ)$$

Question 9

Which of the following best describes the process of finding the x-component of a vector using projection?

Accepted Answer

Draw a line from the tip perpendicular to the x-axis and connect the tail to the intercept.

Question 10

Given vectors $$\vec{A}$$ and $$\vec{B}$$, both originating from the same point, how do you find the resultant vector $$\vec{R}$$ using the parallelogram method?

Accepted Answer

Draw $$\vec{A}$$ and $$\vec{B}$$ tail-to-tail, complete the parallelogram, and $$\vec{R} is $$the diagonal from the common tail.

Question 11

A car travels $$45\,\mathrm{km}$$ east, then $$60\,\mathrm{km} at$$ $$70^\circ$$ south of east. What is the best method to find the resultant displacement?

Accepted Answer

Break each vector into components, sum the components, and use the Pythagorean theorem.

Question 12

If $$\vec{A}$$ has components $$A_x = 9.6\,\mathrm{m}$$ and $$A_y = 7.2\,\mathrm{m}$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{(9.6)^2$ + (7.2)^2$}$$

Question 13

A vector $$\vec{D}$$ points into the third quadrant. Which of the following is true about its components?

Accepted Answer

Both $$D_x$$ and $$D_y$$ are negative.

Question 14

A vector $$\vec{B}$$ has components $$B_x = 8\,\mathrm{m}$$ and $$B_y = -6\,\mathrm{m}. $$What is the direction angle $$	heta ($$measured from the positive x-axis)?

Accepted Answer

$$	heta = \arctan\left(\frac{-6}{8}ight)$$

Question 15

A pilot flies $$62\,\mathrm{km} at$$ $$72.8^\circ$$ north of east. What is the northward (y) component of the displacement?

Accepted Answer

$$62 	imes \sin(72.8^\circ)$$

Question 16

Which of the following is the correct formula for the resultant $$\vec{R} of $$two perpendicular vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

$$R = \sqrt{A^2$ + B^2$}$$

Question 17

A vector $$\vec{A}$$ has a magnitude of $$10\,\mathrm{m}$$ and points $$30^\circ$$ above the x-axis. What is the y-component $$A_y$$?

Accepted Answer

$$10 	imes \sin(30^\circ)$$

Models, Measurements, and Vectors: Foundations for Physics with Calculus

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