Question

The sine of the angle between the vectors $\overrightarrow{a}=3\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-2\hat{j}+\hat{k}$ is

• A. $\sqrt{\frac{74}{99}}$
• B. $\sqrt{\frac{25}{99}}$
• C. $\sqrt{\frac{37}{99}}$
• D. $\frac{5}{\sqrt{41}}$

Answer 1

The correct option is A $\sqrt{\frac{74}{99}}$

Find Cross product of the vectors

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 1& 1\\ 2& -2& 1\end{array}\right|=3\hat{i}-\hat{j}-8\hat{k}$

$\Rightarrow |\overrightarrow{a}\times \overrightarrow{b}|=|3\hat{i}-\hat{j}-8\hat{k}|=\sqrt{74}$

Now we know that

$\sin \theta =\frac{|\overrightarrow{a}\times \overrightarrow{b}|}{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}=\frac{\sqrt{74}}{\sqrt{11}\sqrt{9}}=\sqrt{\frac{74}{99}}$