Question

If $| \mathbf { a } | = 3 , | \mathbf { b } | = 4$ then a value of λ 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. $\frac { 4 } { 3 }$

Answer 1

Answer: (B)

$\frac { 3 } { 4 }$

Answer 2

Since $\mathbf { a } + \lambda \mathbf { b }$ is perpendicular to $\mathbf { a } - \lambda \mathbf { b }$ , then their product will be zero.

So, $( \mathbf { a } + \lambda \mathbf { b } ) \cdot ( \mathbf { a } - \lambda \mathbf { b } ) = 0$ ⇒ $| \mathbf { a } | ^ { 2 } - \lambda ^ { 2 } | \mathbf { b } | ^ { 2 } = 0$

or $\lambda ^ { 2 } = \frac { | \mathbf { a } | ^ { 2 } } { | \mathbf { b } | ^ { 2 } } \Rightarrow \lambda ^ { 2 } = \frac { 9 } { 16 }$ or $\lambda = \pm \frac { 3 } { 4 }$ , $[ \because | \mathbf { a } | = 3 , | \mathbf { b } | = 4 ]$ .