Question

A unit vector $\overline{r}$ makes an angle of measures $\frac{\pi }{3}$ and $\frac{\pi }{4}$ with the positive direction of X-axis and Y-axis respectively. If $\overline{r}$ makes an obtuse angle with the positive direction of Z-axis, then find the measure of the angle.

Answer 1

${\cos }^2\left(\alpha \right)+{\cos }^2\left(\beta \right)+{\cos }^2\left(\gamma \right)=1$

${\cos }^2\left(\frac{\pi }{3}\right)+{\cos }^2\left(\frac{\pi }{4}\right)+{\cos }^2\left(\gamma \right)=1$

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

$\frac{1}{4}+\frac{1}{2}+{\cos }^2\left(\gamma \right)=1$

${\cos }^2\left(\gamma \right)=1-\frac{3}{4}$

${\cos }^2\left(\gamma \right)=\frac{1}{4}$

$\cos \gamma =\pm \frac{1}{2}$

as $\gamma$ is obtuse angle it lies in $2^{nd}$ quadrant

$r={120}^o$.