Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

$$\begin{array}{rl}\cos \frac{\pi }{3}& =\left|\frac{4(3)+(-3)(4)+5\mathrm{k}}{\sqrt{4^2+(-3{)}^2+5^2}\sqrt{3^2+4^2+{\mathrm{k}}^2}}\right|\\ \therefore \frac{1}{2}& =\left|\frac{5\mathrm{k}}{5\sqrt{2}\sqrt{25+{\mathrm{k}}^2}}\right|\end{array}$$ On squaring both side we get $$\begin{array}{rl}& \frac{1}{4}=\frac{25{\mathrm{k}}^2}{50\times \left(25+{\mathrm{k}}^2\right)}\\ \therefore & 100{\mathrm{k}}^2=50\left(25+{\mathrm{k}}^2\right)\Rightarrow {\mathrm{k}}^2=25\Rightarrow \mathrm{k}=\pm 5\end{array}$$