Question

A line $AB$ in three-dimensional space makes angles $45°and120°$ with the positive x-axis and the positive y-axis respectively. If $AB$ makes an acute angle $\theta$ with the positive respectively. If $AB$ makes an acute angle $\theta$ with the positive z-axis, then $\theta$ equals

• A. $30°$
• B. $45°$
• C. $60°$
• D. $75°$

Answer 1

The correct option is C

$60°$

Explanation for the correct option:

Finding the value :

$\begin{array}{rcl}Let,\alpha & =& 45°\\ \beta & =& 120°\end{array}$

Since AB makes an acute angle $\theta$ with the positive z-axis,

${\cos }^2\left(\alpha \right)+{\cos }^2\left(\beta \right)+{\cos }^2\left(\theta \right)=0$

$$\begin{gathered} \Rightarrow {\cos }^2\left(45\right)+{\cos }^2\left(120\right)+{\cos }^2\left(\theta \right)=1 \\ \Rightarrow \frac{1}{2}+\frac{1}{4}+{\cos }^2\left(\theta \right)=1 \\ \Rightarrow {\cos }^2\left(\theta \right)=1-\frac{1}{2}-\frac{1}{4} \\ \Rightarrow {\cos }^2\left(\theta \right)=\frac{1}{4} \\ \Rightarrow \theta =60 \end{gathered}$$

Hence, $\theta$=$60°$

Hence, the correct answer is option (C).