Given three vectors $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{C}$ shown in Figure 1, which of the following statements is true about their vector sum $\overrightarrow{R}$ = $\overrightarrow{A}$ + $\overrightarrow{B}$ + $\overrightarrow{C}$?

• A. The vector sum $\overrightarrow{R}$ is always zero regardless of the arrangement of the vectors.
• B. The direction of $\overrightarrow{R}$ depends on both the magnitudes and directions of the three vectors.
• C. The magnitude of $\overrightarrow{R}$ is always equal to the sum of the magnitudes of the three vectors.
• D. The magnitude of $\overrightarrow{R}$ is always less than the magnitude of any individual vector.

Answer: (B)

Verified step by step guidance

1. Recall that the vector sum \( \vec{R} = \vec{A} + \vec{B} + \vec{C} \) depends on both the magnitudes and directions of the individual vectors \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \).
2. Understand that vectors add according to the parallelogram or triangle rule, meaning their directions affect the resultant vector's direction and magnitude.
3. Note that the magnitude of the resultant vector \( |\vec{R}| \) is not simply the sum of the magnitudes \( |\vec{A}| + |\vec{B}| + |\vec{C}| \) unless all vectors point in exactly the same direction.
4. Recognize that the resultant vector \( \vec{R} \) is not always zero; it will be zero only if the vectors form a closed triangle (i.e., they sum to zero), which depends on their directions and magnitudes.
5. Conclude that the direction of \( \vec{R} \) depends on both the magnitudes and directions of \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \), making the statement about the direction being dependent on both correct.