Question

The sine of the angle between the vectors $\overrightarrow{i}-2\overrightarrow{j}+3\overrightarrow{k}$ and $2\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ is

• A. $\frac{5}{2\sqrt7}$
• B. $\frac{5}{\sqrt7}$
• C. $\frac{3}{\sqrt14}$
• D. $\frac{5}{21}$

Answer 1

Answer: (A)

$cos\,\theta=\frac{\vec{a}.\vec{b}}{\left|\vec{a}\right|\left|\vec{b}\right|}$
$=\frac{2-2+3}{\sqrt{1+4+9}\sqrt{4+1+1}}$
$=\frac{3}{\sqrt{84}}=\frac{\sqrt{3}}{\sqrt{28}}$
$\therefore sin\,\theta=\sqrt{1-cos^{2}\,\theta}$
$=\sqrt{1-\frac{3}{28}}=\frac{5}{\sqrt{28}}=\frac{5}{2\sqrt{7}}$