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  4. The sine of the angle between the vectors i-2j+3k and 2i+j+k is

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Q. 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

Vector Algebra

Solution:

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