Question

If θ is the angle between the vectors $2\hat{i}-2\hat{j}+4\hat{k}\mathrm{and}3\hat{i}+\hat{j}+2\hat{k},$ then sin θ =
(a) $\frac{2}{3}$

(b) $\frac{2}{\sqrt{7}}$

(c) $\frac{\sqrt{2}}{7}$

(d) $\sqrt{\frac{2}{7}}$

Answer 1

(b) $\frac{2}{\sqrt{7}}$

$$\begin{gathered} \text{Let:} \\ \overrightarrow{a}\text{=2}\hat{i}\text{-2}\hat{j}\text{+4}\hat{k} \\ \overrightarrow{b}\text{=3}\hat{i}\text{+}\hat{j}\text{+2}\hat{k} \\ \left|\overrightarrow{a}\right|=\sqrt{2^2+{\left(-2\right)}^2+4^2} \\ =\sqrt{4+4+16} \\ =\sqrt{24} \\ =2\sqrt{6} \\ \left|\overrightarrow{b}\right|=\sqrt{3^2+1^2+2^2} \\ =\sqrt{9+1+4} \\ =\sqrt{14} \\ \overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -2& 4\\ 3& 1& 2\end{array}\right| \\ =-8\hat{i}+8\hat{j}+8\hat{k} \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{64+64+64} \\ =\sqrt{192} \\ =8\sqrt{3} \\ \text{Let}\mathit{\text{θ}}\text{be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b.} \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \\ \Rightarrow 8\sqrt{3}=\left(2\sqrt{6}\right)\left(\sqrt{14}\right)\sin \theta \\ \Rightarrow \sin \theta =\frac{8\sqrt{3}}{4\sqrt{21}} \\ =\frac{2}{\sqrt{7}} \\ \Rightarrow \theta ={\sin }^{-1}\left(\frac{2}{\sqrt{7}}\right) \end{gathered}$$