Question

If \( |\mathbf{a}| = 3 \), \( |\mathbf{b}| = 4 \), then the value of \( \lambda \) for which \( \mathbf{a} + \lambda \mathbf{b} \) is perpendicular to \( \mathbf{a} - \lambda \mathbf{b} \) is:

• A. \( \frac{9}{16} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{3}{2} \)
• D. \( \frac{4}{3} \)

Hint:

For perpendicularity, the dot product must be zero: \( \mathbf{u} \cdot \mathbf{v} = 0 \).

Answer 1

The Correct Option is B

Step 1: Using the perpendicularity condition.
Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular if: \[ \mathbf{u} \cdot \mathbf{v} = 0 \] Setting \( \mathbf{u} = \mathbf{a} + \lambda \mathbf{b} \) and \( \mathbf{v} = \mathbf{a} - \lambda \mathbf{b} \), we compute their dot product: \[ (\mathbf{a} + \lambda \mathbf{b}) \cdot (\mathbf{a} - \lambda \mathbf{b}) = 0 \] Step 2: Expanding the dot product.
Using distributive property, \[ \mathbf{a} \cdot \mathbf{a} - \lambda \mathbf{a} \cdot \mathbf{b} + \lambda \mathbf{b} \cdot \mathbf{a} - \lambda^2 \mathbf{b} \cdot \mathbf{b} = 0 \] Step 3: Substituting given values.
\[ |\mathbf{a}|^2 - \lambda^2 |\mathbf{b}|^2 = 0 \] Since \( |\mathbf{a}| = 3 \) and \( |\mathbf{b}| = 4 \), \[ 9 - \lambda^2 (16) = 0 \] Step 4: Solving for \( \lambda \).
\[ \lambda^2 = \frac{9}{16} \] \[ \lambda = \frac{3}{4} \] Thus, the correct answer is (B).