Question

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors such that $\overrightarrow{a}$ is perpendicular to the plane of $\overrightarrow{b}$ and $\overrightarrow{c}$.If the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi }{3}$, then $|\overrightarrow{a}\times \overrightarrow{b}-\overrightarrow{a}\times \overrightarrow{c}|=$

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

Answer 1

The correct option is C $1$

Here $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{a}.\overrightarrow{c}=0$
$\overrightarrow{b}.\overrightarrow{c}=\cos \frac{\pi }{3}=\frac{1}{2}$
$|\overrightarrow{a}\times \overrightarrow{b}|=|\overrightarrow{a}\times \overrightarrow{c}|=1$ from above
${|\overrightarrow{a}\times \overrightarrow{b}-\overrightarrow{a}\times \overrightarrow{c}|}^2={|\overrightarrow{a}\times \overrightarrow{b}|}^2+{|\overrightarrow{a}\times \overrightarrow{c}|}^2-2(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{c})$
$=1+1-2\left|\begin{array}{cc}1& 0\\ 0& \frac{1}{2}\end{array}\right|$
$=1+1-2\times \frac{1}{2}$
$=1+1-1=1$