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

Given three vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) with magnitudes \( |\mathbf{a}| = 2 \), \( |\mathbf{b}| = 3 \), and \( |\mathbf{c}| = 4 \). The following perpendicularity conditions are provided:1) \( \mathbf{a} \perp (\mathbf{b} + \mathbf{c}) \)2) \( \mathbf{b} \perp (\mathbf{c} + \mathbf{a}) \)3) \( \mathbf{c} \perp (\mathbf{a} + \mathbf{b}) \).Determine the magnitude of the vector sum \( |\mathbf{a} + \mathbf{b} + \mathbf{c}| \).
Step 1: Apply dot product for perpendicularity.Perpendicular vectors have a dot product of zero.- \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 0 \)- \( \mathbf{b} \cdot (\mathbf{c} + \mathbf{a}) = 0 \)- \( \mathbf{c} \cdot (\mathbf{a} + \mathbf{b}) = 0 \)
Step 2: Expand the dot products.Distributing the dot product yields:- \( \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} = 0 \)- \( \mathbf{b} \cdot \mathbf{c} + \mathbf{b} \cdot \mathbf{a} = 0 \)- \( \mathbf{c} \cdot \mathbf{a} + \mathbf{c} \cdot \mathbf{b} = 0 \)This system of equations implies that \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are mutually orthogonal.
Step 3: Calculate the magnitude of the sum.For orthogonal vectors, the magnitude of their sum is found using the generalized Pythagorean theorem:\[|\mathbf{a} + \mathbf{b} + \mathbf{c}| = \sqrt{|\mathbf{a}|^2 + |\mathbf{b}|^2 + |\mathbf{c}|^2}\]Substituting the given magnitudes:\[|\mathbf{a} + \mathbf{b} + \mathbf{c}| = \sqrt{2^2 + 3^2 + 4^2}\]\[|\mathbf{a} + \mathbf{b} + \mathbf{c}| = \sqrt{4 + 9 + 16} = \sqrt{29}\]The magnitude of \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) is \( \boxed{\sqrt{29}} \).