Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

$\cos \alpha ,\cos \beta ,\cos \gamma$ i.e. $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{\sqrt{2}}$. Let $\alpha ,\beta$ and $\gamma$ be the angles made by the line segment OP with X-axis, Y-axis and Z-axis, respectively. Given: $\alpha =\frac{\pi }{4}$ and $\beta =\frac{\pi }{3}$ We know that, ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$ $\begin{array}{l}\therefore {\cos }^2\frac{\pi }{4}+{\cos }^2\frac{\pi }{3}+{\cos }^2\gamma =1\\ \Rightarrow {\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2+{\cos }^2\gamma =1\\ \Rightarrow \frac{1}{2}+\frac{1}{4}+{\cos }^2\gamma =1\\ \Rightarrow {\cos }^2\gamma =\frac{1}{4}\\ \Rightarrow \cos \gamma =\frac{1}{\sqrt{2}}\\ \therefore \gamma =\frac{\pi }{4}\end{array}$ Hence, direction cosines are $\cos \alpha ,\cos \beta ,\cos \gamma$ i.e. $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{\sqrt{2}}$.