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Question
If θ is the angle between the vectors \[2 \hat{ i } - 2 \hat{ j} + 4 \hat{ k } \text{ and } 3 \hat{ i } + \hat { j } + 2 \hat{ k } ,\] then sin θ =
Options
\[\frac{2}{3}\]
\[\frac{2}{\sqrt{7}}\]
\[\frac{\sqrt{2}}{7}\]
\[\sqrt{\frac{2}{7}}\]
Solution
\[\frac{2}{\sqrt{7}}\]
\[\text{ Let } :\]
\[ \vec{a} =2 \hat{ i } -2 \hat{ j } +4 \hat{ k } \]
\[ \vec{b} =3 \hat{ i } + \hat{ j } +2 \hat{ k } \]
\[\left| \vec{a} \right| = \sqrt{2^2 + \left( - 2 \right)^2 + 4^2}\]
\[ = \sqrt{4 + 4 + 16}\]
\[ = \sqrt{24}\]
\[ = 2\sqrt{6}\]
\[ \left| \vec{b} \right| = \sqrt{3^2 + 1^2 + 2^2}\]
\[ = \sqrt{9 + 1 + 4}\]
\[ = \sqrt{14}\]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & - 2 & 4 \\ 3 & 1 & 2\end{vmatrix}\]
\[ = - 8 \hat{ i } + 8 \hat{ j } + 8 \hat{ k } \]
\[\left| \vec{a} \times \vec{b} \right| = \sqrt{64 + 64 + 64}\]
\[ = \sqrt{192}\]
\[ = 8 \sqrt{3}\]
\[\text{ Let } \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b .} \]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta\]
\[ \Rightarrow 8 \sqrt{3} = \left( 2\sqrt{6} \right)\left( \sqrt{14} \right) \sin \theta\]
\[ \Rightarrow \sin \theta = \frac{8 \sqrt{3}}{4\sqrt{21}}\]
\[ = \frac{2}{\sqrt{7}}\]
\[ \Rightarrow \theta = \sin^{- 1} \left( \frac{2}{\sqrt{7}} \right)\]
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