Question

If sine of the angle between the vectors $\overrightarrow{a}=3i+4k$ & $\overrightarrow{b}=i+3j$ is $\frac{m}{\sqrt{10}}$ . Find m

Answer 1

$\overline{a}\times \overline{b}=\left|\begin{array}{ccc}\stackrel{\wedge }{i}& \stackrel{\wedge }{j}& \stackrel{\wedge }{k}\\ 3& 0& 4\\ 1& 3& 0\end{array}\right|$

$⟹\stackrel{\wedge }{i}(0-12)-\stackrel{\wedge }{j}(0-4)+\stackrel{\wedge }{k}\left(9\right)$

$⟹-12\stackrel{\wedge }{i}+4\stackrel{\wedge }{j}+9\stackrel{\wedge }{k}$

$⟹|\overline{a}\times \overline{b}|=\sqrt{144+16+81}=\sqrt{241}$

$\left|\overline{a}\right|=\sqrt{3^2+4^2}=5$

$\left|\overline{b}\right|=\sqrt{1^2+3^2}=\sqrt{10}$

$\therefore sin\theta =\frac{|\overline{a}\times \overline{b}|}{\left|\overline{a}\right|.\left|\overline{b}\right|}=\frac{\sqrt{241}}{5\sqrt{10}}$

$⟹m=\frac{\sqrt{241}}{5}$