Question

Six vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}, \mathbf{e}, \mathbf{f} \) have the magnitudes and directions indicated in the figure. Which of the following statements is true?

• A. \( \mathbf{b} + \mathbf{e} = \mathbf{f} \)
• B. \( \mathbf{b} + \mathbf{c} = \mathbf{f} \)
• C. \( \mathbf{d} + \mathbf{c} = \mathbf{f} \)
• D. \( \mathbf{d} + \mathbf{e} = \mathbf{f} \)

Hint:

When adding vectors, use the head-to-tail method to find the resultant vector. Ensure the magnitudes and directions align correctly.

Answer 1

The Correct Option is D

To determine the correct answer, we will analyze the vectors based on their directions and magnitudes as depicted in the image. We will examine how the vectors relate to each other, considering their orientations, and utilize vector addition to identify the correct statement.
Step 1: Vector Addition Principle Vector addition is commutative; the order of addition does not affect the result. The sum of two vectors can be found geometrically using the head-to-tail method.
Step 2: Statement Verification 1. Statement (A): \( \mathbf{b} + \mathbf{e} = \mathbf{f} \) We check if \( \mathbf{b} \) plus \( \mathbf{e} \) equals \( \mathbf{f} \), ensuring consistent direction and magnitude. 2. Statement (B): \( \mathbf{b} + \mathbf{c} = \mathbf{f} \) We verify if adding \( \mathbf{b} \) and \( \mathbf{c} \) results in \( \mathbf{f} \), using geometric vector addition. 3. Statement (C): \( \mathbf{d} + \mathbf{c} = \mathbf{f} \) We check if adding \( \mathbf{d} \) and \( \mathbf{c} \) results in \( \mathbf{f} \). 4. Statement (D): \( \mathbf{d} + \mathbf{e} = \mathbf{f} \) We verify if the sum of vectors \( \mathbf{d} \) and \( \mathbf{e} \) geometrically equals \( \mathbf{f} \).
Step 3: Conclusion After performing the vector addition and analyzing directions and magnitudes, we conclude that the correct option is: \[\n\boxed{(D) \, \mathbf{d} + \mathbf{e} = \mathbf{f}}\n\] This is the only combination where the vectors align geometrically to produce the correct resulting vector.