Suppose you have four vectors: $A$ points along the positive $x$-axis, $B$ points along the positive $y$-axis, $C$ points along the negative $x$-axis, and $D$ points at $45°$ above the positive $x$-axis. Which pair of vectors, when added together, will result in the largest positive $x$ component?

• A. $A$ and $B$
• B. $B$ and $C$
• C. $A$ and $D$
• D. $C$ and $D$

Answer: (C)

Verified step by step guidance

1. Identify the direction and components of each vector based on the problem description: Vector A points along the positive x-axis, so its x-component is positive and y-component is zero. Vector B points along the positive y-axis, so its x-component is zero and y-component is positive. Vector C points along the negative x-axis, so its x-component is negative and y-component is zero. Vector D points at 45° above the positive x-axis, so both its x and y components are positive and equal in magnitude (since 45° implies equal components).
2. Express the x-components of each vector: For A, x-component is \$A_x = |A|\$; for B, \$B_x = 0\$; for C, \$C_x = -|C|\$; for D, \$D_x = |D| \(\cos\)(45^\(\circ\))\$.
3. For each pair of vectors, calculate the sum of their x-components: For example, for A and B, sum is \$A_x + B_x = |A| + 0 = |A|\$; for B and C, sum is \$0 + (-|C|) = -|C|\$; for A and D, sum is \$|A| + |D| \(\cos\)(45^\(\circ\))\$; for C and D, sum is \$- |C| + |D| \(\cos\)(45^\(\circ\))\$.
4. Compare these sums to determine which pair yields the largest positive x-component. Since \$|A|\$ and \$|D| \(\cos\)(45^\(\circ\))\$ are both positive, their sum is likely larger than the others, especially compared to pairs involving negative x-components.
5. Conclude that the pair with the largest positive x-component is the one whose sum of x-components is greatest and positive, which is the pair A and D.