Suppose you have four vectors of equal magnitude: $\mathrm{Vector} A$ points along the positive $x$-axis, $\mathrm{Vector} B$ points along the positive $y$-axis, $\mathrm{Vector} C$ points at $45°$ above the $x$-axis (in the first quadrant), and $\mathrm{Vector} D$ points at $45°$ below the $x$-axis (in the fourth quadrant). Which pair of vectors, when added, will result in the largest positive $y$ component?

• A. $\mathrm{Vector} C$ and $\mathrm{Vector} D$
• B. $\mathrm{Vector} A$ and $\mathrm{Vector} D$
• C. $\mathrm{Vector} A$ and $\mathrm{Vector} B$
• D. $\mathrm{Vector} B$ and $\mathrm{Vector} C$

Answer: (D)

Verified step by step guidance

1. Identify the direction and components of each vector. Since all vectors have equal magnitude, denote the magnitude as \( M \). Vector \( A \) points along the positive x-axis, so its components are \( (M, 0) \). Vector \( B \) points along the positive y-axis, so its components are \( (0, M) \).
2. For Vector \( C \), which points 45° above the x-axis in the first quadrant, calculate its components using trigonometry: \( C_x = M \cos 45^\circ \) and \( C_y = M \sin 45^\circ \). Since \( \cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \), both components are positive and equal.
3. For Vector \( D \), which points 45° below the x-axis in the fourth quadrant, calculate its components similarly: \( D_x = M \cos (-45^\circ) = M \cos 45^\circ \) (positive) and \( D_y = M \sin (-45^\circ) = -M \sin 45^\circ \) (negative).
4. To find the pair of vectors that results in the largest positive y-component when added, sum the y-components of each possible pair: (A and B), (A and D), (C and D), and (B and C). Remember that Vector \( A \) has zero y-component, Vector \( B \) has positive y-component \( M \), Vector \( C \) has positive y-component \( M \sin 45^\circ \), and Vector \( D \) has negative y-component \( -M \sin 45^\circ \).
5. Compare the sums of the y-components for each pair. The pair with the largest positive sum in the y-direction will be the answer. This involves adding the y-components and identifying which sum is greatest.