Question

If $|\overrightarrow{a}|=3,|\overrightarrow{b}|=4$ then a value of $\lambda$ for which $\overrightarrow{a}+\lambda \overrightarrow{b}$ is perpendicular to $\overrightarrow{a}-\lambda \overrightarrow{b}$ is?

• A. $9/16$
• B. $3/4$
• C. $3/2$
• D. $4/3$

Answer 1

The correct option is B $3/4$

For vctors to be perpendicular:

$\begin{array}{c}\left(\overrightarrow{a}+\lambda \overrightarrow{b}\right)\left(\overrightarrow{a}-\lambda \overrightarrow{b}\right)=0\\ \overrightarrow{a}.\overrightarrow{a}-\lambda \overrightarrow{a}\overrightarrow{b}+\lambda \overrightarrow{b}\overrightarrow{a}-{\lambda }^2\overrightarrow{b}\overrightarrow{b}=0\\ {\left|\overrightarrow{a}\right|}^2-{\lambda }^2{\left|\overrightarrow{b}\right|}^2=0\\ {\left(3\right)}^2-{\lambda }^2{\left(4\right)}^2=0\\ 9-16{\lambda }^2=0\\ 16{\lambda }^2=9\\ \lambda =\frac{9}{16}\\ \lambda =\frac{3}{4}\end{array}$

$B$ is the correct answer