Question

If a line makes \(90^\circ\), \(60^\circ\) and \(30^\circ\) with the \(x\), \(y\) and \(z\)-axes respectively in the positive direction, then find its direction cosines.

Hint:

Direction cosines are l = cos(angle with x-axis), m = cos(angle with y-axis), n = cos(angle with z-axis).

Answer 1

Step 1: Key Formula:
If a line makes angles \(\alpha, \beta, \gamma\) with the positive \(x, y, z\) axes, its direction cosines are \(l = \cos\alpha\), \(m = \cos\beta\), \(n = \cos\gamma\).
Here \(\alpha = 90^\circ\), \(\beta = 60^\circ\), \(\gamma = 30^\circ\).

Step 2: Compute each cosine:
\(\cos 90^\circ = 0\), \(\cos 60^\circ = \dfrac{1}{2}\), \(\cos 30^\circ = \dfrac{\sqrt3}{2}\).
So the direction cosines are:
\[ l = 0, \quad m = \frac{1}{2}, \quad n = \frac{\sqrt3}{2} \]

Step 3: Verify using l^2+m^2+n^2=1:
Check the standard identity that direction cosines must satisfy:
\[ 0^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt3}{2}\right)^2 = 0 + \frac{1}{4} + \frac{3}{4} = 1 \]
The identity holds, so the values are correct.

Final Answer:
The direction cosines of the line are 0, one half, and root 3 by 2. \[ \boxed{l = 0,\ m = \frac{1}{2},\ n = \frac{\sqrt3}{2}} \]