Question

Let \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) be vectors of magnitude 2, 3, and 4 respectively. If: - \( \mathbf{a} \) is perpendicular to \( (\mathbf{b} + \mathbf{c}) \), - \( \mathbf{b} \) is perpendicular to \( (\mathbf{c} + \mathbf{a}) \), - \( \mathbf{c} \) is perpendicular to \( (\mathbf{a} + \mathbf{b}) \), then the magnitude of \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) is equal to:

• A. 29
• B. \( \sqrt{29} \)
• C. 26
• D. \( \sqrt{26} \)

Hint:

When vectors are perpendicular, the magnitude of their sum can be found using the Pythagorean theorem.

Answer 1

The Correct Option is B

We are given three vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) with magnitudes 2, 3, and 4, respectively. We are also given the following conditions: 1) \( \mathbf{a} \) is perpendicular to \( \mathbf{b} + \mathbf{c} \), 2) \( \mathbf{b} \) is perpendicular to \( \mathbf{c} + \mathbf{a} \), 3) \( \mathbf{c} \) is perpendicular to \( \mathbf{a} + \mathbf{b} \). We need to find the magnitude of the vector sum \( \mathbf{a} + \mathbf{b} + \mathbf{c} \).
Step 1: Use the perpendicularity conditions For each pair of vectors, their dot product must be zero because they are perpendicular. - \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 0 \) - \( \mathbf{b} \cdot (\mathbf{c} + \mathbf{a}) = 0 \) - \( \mathbf{c} \cdot (\mathbf{a} + \mathbf{b}) = 0 \)
Step 2: Simplify the equations From the first condition: \[ \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 0 \implies \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} = 0 \] From the second condition: \[ \mathbf{b} \cdot (\mathbf{c} + \mathbf{a}) = 0 \implies \mathbf{b} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{a} = 0 \] From the third condition: \[ \mathbf{c} \cdot (\mathbf{a} + \mathbf{b}) = 0 \implies \mathbf{c} \cdot \mathbf{a} + \mathbf{c} \cdot \mathbf{b} = 0 \] Thus, we have the system of three equations: 1) \( \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} = 0 \) 2) \( \mathbf{b} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{a} = 0 \) 3) \( \mathbf{c} \cdot \mathbf{a} + \mathbf{c} \cdot \mathbf{b} = 0 \) These equations imply that \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are mutually perpendicular to each other.
Step 3: Calculate the magnitude of \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) Since the vectors are perpendicular to each other, the magnitude of their sum is given by the Pythagorean theorem: \[ |\mathbf{a} + \mathbf{b} + \mathbf{c}| = \sqrt{|\mathbf{a}|^2 + |\mathbf{b}|^2 + |\mathbf{c}|^2} \] Substitute the magnitudes of \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \): \[ |\mathbf{a} + \mathbf{b} + \mathbf{c}| = \sqrt{2^2 + 3^2 + 4^2} \] \[ |\mathbf{a} + \mathbf{b} + \mathbf{c}| = \sqrt{4 + 9 + 16} = \sqrt{29} \] Thus, the magnitude of \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) is \( \boxed{\sqrt{29}} \).