Question

A st. line which makes angle of ${60}^{\circ }$ with each of y - and z - axes, is inclined with x - axis at an angle

• A. ${45}^{\circ }$
• B. ${30}^{\circ }$
• C. ${75}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is A ${45}^{\circ }$

As we know in $3D$ the relation between angles made by a vector with $x$ axis $\left(x\right),y$ axis $\left(y\right),z$ axis$\left(z\right)$ is:-

${\cos }^{2}x+{\cos }^{2}y+{\cos }^{2}z=1$

$\Rightarrow {\cos }^2{60}^{\circ }+{\cos }^2{60}^{\circ }+{\cos }^{2}z=1$

$\Rightarrow {\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2+{\cos }^{2}z=1$

$\Rightarrow \frac{1}{4}+\frac{1}{4}+{\cos }^{2}z=1$

$\Rightarrow \frac{2}{4}+{\cos }^{2}z=1$

$\Rightarrow \frac{1}{2}+{\cos }^{2}z=1$

$\Rightarrow {\cos }^{2}z=1-\frac{1}{2}=\frac{1}{2}$

$\Rightarrow \cos z=\frac{1}{\sqrt{2}}$

$\therefore z={45}^{\circ }$