Question

A line $AB$ in three-dimensional space makes angles ${45}^o$ and ${120}^o$ with the positive $x$-axis and the positive $y$-axis respectively. lf $AB$ makes an acute angle $e$ with the positive $z$-axis, then $e$ equals

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

Answer 1

Answer: (B)

${60}^o$

The angle made by line $AB$ with $x$ axis is ${45}^0$

The angle made by line $AB$ with $y$ axis is ${120}^0$

Let the angle made by line $AB$ with $z$ axis is $\theta$

We have ${\cos }^2\left(45\right)+{\cos }^2\left(120\right)+{\cos }^2\theta =1$

$\Rightarrow co{s}^2\theta =1-\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$

Given that $\theta$ is accute

So we get $cos\theta =\frac{1}{2}$

$\Rightarrow \theta ={60}^0$

Therefore option $B$ is correct