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 =

 

पर्याय

• \[\frac{2}{3}\]

• \[\frac{2}{\sqrt{7}}\]

• \[\frac{\sqrt{2}}{7}\]

• \[\sqrt{\frac{2}{7}}\] 

Answer 1

\[\frac{2}{\sqrt{7}}\] 

\[\text{ Let } :\]

\[ \vec{a} =2 \hat{ i }  -2 \hat{ j }  +4 \hat{ k }  \]

\[ \vec{b} =3 \hat{ i } + \hat{ j }  +2 \hat{ k }  \]

\[\left| \vec{a} \right| = \sqrt{2^2 + \left( - 2 \right)^2 + 4^2}\]

\[ = \sqrt{4 + 4 + 16}\]

\[ = \sqrt{24}\]

\[ = 2\sqrt{6}\]

\[ \left| \vec{b} \right| = \sqrt{3^2 + 1^2 + 2^2}\]

\[ = \sqrt{9 + 1 + 4}\]

\[ = \sqrt{14}\]

\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i }  & \hat{ j }  & \hat{ k }  \\ 2 & - 2 & 4 \\ 3 & 1 & 2\end{vmatrix}\]

\[ = - 8 \hat{ i } + 8 \hat{ j }  + 8 \hat{ k }  \]

\[\left| \vec{a} \times \vec{b} \right| = \sqrt{64 + 64 + 64}\]

\[ = \sqrt{192}\]

\[ = 8 \sqrt{3}\]

\[\text{ Let }  \theta \text{ be the angle between } \vec{a} \text{ and } \vec{b .} \]

\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \sin \theta\]

\[ \Rightarrow 8 \sqrt{3} = \left( 2\sqrt{6} \right)\left( \sqrt{14} \right) \sin \theta\]

\[ \Rightarrow \sin \theta = \frac{8 \sqrt{3}}{4\sqrt{21}}\]

\[ = \frac{2}{\sqrt{7}}\]

\[ \Rightarrow \theta = \sin^{- 1} \left( \frac{2}{\sqrt{7}} \right)\]