Question

If $q$ is the angle between the vectors $2\hat{i}-2\hat{j}+4\hat{k}$ and $3\hat{i}+\hat{j}+2\hat{k}$, then $\sin q=$

• A. $\frac{2}{3}$
• B. $\frac{2}{\sqrt{7}}$
• C. $\frac{\sqrt{2}}{7}$
• D. $\sqrt{\frac{2}{7}}$

Answer 1

Answer: (B)

$\frac{2}{\sqrt{7}}$

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -1& 4\\ 3& 1& 2\end{array}\right|=(-8\hat{i}+8\hat{j}+8\hat{k})$
$|\overrightarrow{a}\times \overrightarrow{b}|=8\sqrt{3}$
$|\overrightarrow{a}|=2\sqrt{6}|\overrightarrow{b}|=\sqrt{14}$
$|\overrightarrow{a}\times \overrightarrow{b}|=|\overrightarrow{a}||\overrightarrow{b}|\sin q$
$8\sqrt{3}=2\sqrt{6}\times \sqrt{14}\times \sin q$
$\frac{2}{\sqrt{7}}=\sin q$