Question

If a vector $\overline{x}$ makes angle with measure $\frac{\pi }{3}$ and $\frac{2\pi }{3}$ with $x$-axis and $y$-axis respectively, then find the measure of the anlge made by $\overline{x}$ with z-axis.

Answer 1

given that if a vector makes angle $\alpha$ with $x$ axis $\beta$ with $y$ axis and $\gamma$ with $z$ axis respectively then${cos}^2\left(\alpha \right)+{cos}^2\left(\beta \right)+{cos}^2\left(\gamma \right)=1$

therefore by substituting $\alpha =\frac{\pi }{3}and\beta =\frac{2\pi }{3}$ we get

${cos}^2\left(\gamma \right)=\frac{1}{\sqrt{2}}$

$\gamma$=${45}^0$