Question

If a line makes angles $45^\circ$ and $60^\circ$ with the positive $x$ and $y$ axes respectively, then the acute angle it makes with the $z$-axis is:

• A. $60^\circ$
• B. $30^\circ$
• C. $45^\circ$
• D. $90^\circ$

Hint:

Remember that the sum of squares of direction cosines is always 1. This is one of the most fundamental relations in 3D geometry!

Answer 1

The Correct Option is A

Step 1: Concept
If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then their direction cosines $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$ satisfy the identity $l^2 + m^2 + n^2 = 1 \implies \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$.

Step 2: Meaning
We are given $\alpha = 45^\circ$ and $\beta = 60^\circ$. We need to find the acute angle $\gamma$.

Step 3: Analysis
Substituting the given angles: \[ \cos^2 45^\circ + \cos^2 60^\circ + \cos^2 \gamma = 1 \] \[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + \cos^2 \gamma = 1 \] \[ \frac{1}{2} + \frac{1}{4} + \cos^2 \gamma = 1 \] \[ \frac{3}{4} + \cos^2 \gamma = 1 \implies \cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4} \] Since we need the acute angle, we take the positive root: \[ \cos\gamma = \frac{1}{2} \implies \gamma = 60^\circ \]

Step 4: Conclusion
The acute angle the line makes with the $z$-axis is $60^\circ$. Final Answer: (A)