Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are vectors such that $\overrightarrow{a}.\overrightarrow{b}=0$ and $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c},$ then

• A. ${\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2={\left|c\right|}^2$
• B. ${\left|\overrightarrow{a}\right|}^2={\left|\overrightarrow{b}\right|}^2+{\left|c\right|}^2$
• C. ${\left|\overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2+{\left|c\right|}^2$
• D. None of these

Answer 1

Answer: (A)

${\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2={\left|c\right|}^2$

We have $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$,

Squaring, we get

$\Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}={\left|\overrightarrow{c}\right|}^2\Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+0={\left|\overrightarrow{c}\right|}^2\therefore {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2={\left|\overrightarrow{c}\right|}^2$