Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ be two perpendicular unit vectors such that $\overrightarrow{x}=\overrightarrow{b}-(\overrightarrow{a}\times \overrightarrow{x})$, then $\left|\overrightarrow{x}\right|$ is equal to

• A. $1$
• B. $\sqrt{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\sqrt{3}$

Answer 1

The correct option is A $1$

Two vectors are said to be perpendicular if $\overrightarrow{a}.\overrightarrow{b}=0$

Consider $\overrightarrow{x}=\overrightarrow{b}-(\overrightarrow{a}\times \overrightarrow{x})$

$\overrightarrow{x}.\overrightarrow{x}=\overrightarrow{b}.\overrightarrow{x}-(\overrightarrow{a}\times \overrightarrow{x}).\overrightarrow{x}=0$

$\Rightarrow {\overrightarrow{x}}^2=\overrightarrow{b}.\overrightarrow{x}$

Given $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors

$\therefore \left|a\right|=1,\left|b\right|=1$

$\Rightarrow {\left|\overrightarrow{x}\right|}^2=|\overrightarrow{b}.\overrightarrow{x}|$

$\Rightarrow {\left|\overrightarrow{x}\right|}^2=\left|\overrightarrow{b}\right|.\left|\overrightarrow{x}\right|$

$\Rightarrow {\left|\overrightarrow{x}\right|}^2=1.\left|\overrightarrow{x}\right|$

$\Rightarrow \left|\overrightarrow{x}\right|=1$