Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors then the value of $\left(\overrightarrow{a}+\overrightarrow{b}\right)\times \left(\overrightarrow{a}-\overrightarrow{b}\right)$ is:

Answer 1

Let us suppose the angle (defined from a to b clockwise between the vectors is$\theta ,0\le \theta \le 2\pi \theta ,0\le \theta \le 2\pi .$Then using the parallelogram rule for vector addition,and the law of cosines

${|a+b|}^2={|a|}^2+{|b|}^2-2|a||b|cos\theta$

Similarly, recognizing that the angle between $a$ and $-b$ is $\pi -\theta$,we get

${|a-b|}^2={|a|}^2+{|b|}^2-2|a||b|cos(\pi -\theta )={|a|}^2+{|b|}^2+2|a||b|cos\theta .$

As,$|a+b|=|a-b|,$

${|a|}^2+{|b|}^2+2|a||b|cos\theta ={|a|}^2+{|b|}^2+2|a||b|cos\theta$

$cos\theta =0$

$\theta =\frac{\pi }{2}or\frac{3\pi }{2}.$