FIGURE P3.26 shows vectors A and B. Find D = 2A ＋B Write your answer in component form.

검증된 단계별 안내

Step 1: Analyze the given vectors A and B in the diagram. Vector A has a magnitude of 5 m and is directed at an angle of 20° above the negative x-axis. Vector B has a magnitude of 3 m and is directed at an angle of 20° below the negative x-axis.

Step 2: Break down each vector into its components. For vector A, calculate the x-component as \( A_x = -A \cos(20°) \) and the y-component as \( A_y = A \sin(20°) \). Similarly, for vector B, calculate the x-component as \( B_x = -B \cos(20°) \) and the y-component as \( B_y = -B \sin(20°) \).

Step 3: Multiply vector A by 2 to account for the scalar multiplication in \( D = 2A + B \). The new components of 2A will be \( (2A_x, 2A_y) \).

Step 4: Add the components of 2A and B to find the components of vector D. Use \( D_x = 2A_x + B_x \) for the x-component and \( D_y = 2A_y + B_y \) for the y-component.

Step 5: Write the final result for vector D in component form as \( D = (D_x, D_y) \).

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주요 개념

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Vector Addition

Vector addition involves combining two or more vectors to determine a resultant vector. This is done by adding the corresponding components of the vectors. In this case, the vectors A and B must be expressed in their component forms (x and y) before performing the addition to find the resultant vector D.

Component Form of Vectors

The component form of a vector expresses it in terms of its horizontal (x) and vertical (y) components. For a vector with magnitude and angle, the components can be calculated using trigonometric functions: Ax = A * cos(θ) and Ay = A * sin(θ). This allows for easier manipulation and addition of vectors.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In vector analysis, these functions are used to resolve vectors into their components. For example, the angle given in the problem (20 degrees) will be used to calculate the x and y components of vectors A and B, which are essential for finding the resultant vector D.