Question

If $\overline{a}$ and $\overline{b}$ are two $\perp$ vectors, then out of the following four statements
$1){(\overline{a}+\overline{b})}^2={\left(\overline{a}\right)}^2+{\left(\overline{b}\right)}^2$
$2){(\overline{a}-\overline{b})}^2={\left(\overline{a}\right)}^2+{\left(\overline{b}\right)}^2$
$3){|\overline{a}-\overline{b}|}^2={\left|\overline{a}\right|}^2+{\left|\overline{b}\right|}^2$
$4){(\overline{a}+\overline{b})}^2={(\overline{a}-\overline{b})}^2$

• A. Only one is correct
• B. Only two are correct
• C. Only three are correct
• D. All the four are correct

Answer 1

The correct option is D All the four are correct

We know that

$(\overline{a}+\overline{b}{)}^2=(\overline{a}{)}^2+\overline{a}.\overline{b}+\overline{b}.\overline{a}+(\overline{b}{)}^2$
$(\overline{a}-\overline{b}{)}^2=(\overline{a}{)}^2-\overline{a}.\overline{b}-\overline{b}.\overline{a}+(\overline{b}{)}^2$
Now since vectors are perpendicular

$\therefore \overline{a}.\overline{b}=0$,

We get the form given in four options and hence all the options are correct