Question

If ${\displaystyle \hat{a}}$ is unit vector and ${\displaystyle (2\overrightarrow{x}-3\hat{a})\cdot (2\overrightarrow{x}+3\hat{a})}$ = 91, find the value of ${\displaystyle \left|\overrightarrow{x}\right|}$.

Answer 1

∵ ${\displaystyle \hat{a}}$ is a unit vector.

∴ ${\displaystyle \left|\hat{a}\right|}$ = 1

Given ${\displaystyle (2\overrightarrow{x}-3\hat{a})\cdot (2\overrightarrow{x}+3\hat{a})}$ = 91

${\displaystyle \Rightarrow 4\overrightarrow{x}\cdot \overrightarrow{x}+6\overrightarrow{x}\cdot \hat{a}-6\hat{a}\cdot \overrightarrow{x}-9\hat{a}\cdot \hat{a}}$ = 91

${\displaystyle \Rightarrow 4{\left|\overrightarrow{x}\right|}^2-9{\left|\hat{a}\right|}^2}$ = 91   ...${\displaystyle [\because \overrightarrow{x}\cdot \hat{a}=\hat{a}\cdot \overrightarrow{x}]}$

${\displaystyle \Rightarrow 4{\left|\overrightarrow{x}\right|}^2-9\times 1}$ = 91

${\displaystyle 4{\left|\overrightarrow{x}\right|}^2}$ = 91 + 9 = 100

${\displaystyle {\left|\overrightarrow{x}\right|}^2=\frac{100}{4}}$ = 25

∴ ${\displaystyle \left|\overrightarrow{x}\right|=\sqrt{25}}$ = 5