Question

Let \( \vec{a} + \vec{b} = \lambda \hat{i} + 16\hat{j} - 18\hat{k} \) and \( \vec{a} - \vec{b} = 2\hat{i} + 8\hat{j} + \lambda \hat{k} \). If \( \vec{a} + \vec{b} \) is perpendicular to \( \vec{a} - \vec{b} \), then \( |\vec{a}| = \)

• A. \(5\sqrt{13} \)
• B. \( \sqrt{174} \)
• C. \( \sqrt{184} \)
• D. \( 13\sqrt{5} \)
• E. \( \sqrt{194} \)

Hint:

Use \( 2\vec{a} = (\vec{a}+\vec{b}) + (\vec{a}-\vec{b}) \) to quickly extract vectors.

Answer 1

Answer: (E)

Concept: Perpendicular vectors: \[ (\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b}) = 0 \]

Step 1: Apply dot product.
\[ (\lambda,16,-18) \cdot (2,8,\lambda) = 0 \] \[ 2\lambda + 128 - 18\lambda = 0 \Rightarrow -16\lambda + 128 = 0 \Rightarrow \lambda = 8 \]

Step 2: Find \( \vec{a} \).
\[ 2\vec{a} = (\vec{a}+\vec{b}) + (\vec{a}-\vec{b}) \] \[ = (8,16,-18) + (2,8,8) = (10,24,-10) \] \[ \vec{a} = (5,12,-5) \]

Step 3: Find magnitude.
\[ |\vec{a}| = \sqrt{25 + 144 + 25} = \sqrt{194} \]