Mathematics - Vector Question with Solution | TestHub

Given: $\left|\overrightarrow{b}\right|=1, \left|\overrightarrow{b}\times \overrightarrow{a}\right|=2$

Now, solving ${\left|\overrightarrow{b}\times \overrightarrow{a}-\overrightarrow{b}\right|}^2$,

$\Rightarrow {\left|\overrightarrow{b}\times \overrightarrow{a}-\overrightarrow{b}\right|}^2={\left|\overrightarrow{b}\times \overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2-2\left(\overrightarrow{b}\times \overrightarrow{a}\right)·\overrightarrow{b}$

Now, we know that, $\left(\overrightarrow{b}\times \overrightarrow{a}\right)·\overrightarrow{b}=\left(\overrightarrow{b}\times \overrightarrow{a}\right)·\overrightarrow{a}=0$

$\Rightarrow {\left|\overrightarrow{b}\times \overrightarrow{a}-\overrightarrow{b}\right|}^2=2^2+1^2-2\times 0$

$\Rightarrow {\left|\overrightarrow{b}\times \overrightarrow{a}-\overrightarrow{b}\right|}^2=5$