Question

If a unit vector makes angles \(\frac{π}{4}\) with \(\hat{i}\), \(\frac{π}{3}\) with \(\hat{j}\) and \(θ\in (0,π)\) with \(\hat{k}\), then a value of \(θ\) is equal to...

• A. \(\frac{π}{3},\frac{2π}{3}\)
• B. \(\frac{π}{6},\frac{5π}{6}\)
• C. \(\frac{π}{4},\frac{5π}{4}\)
• D. \(\frac{5π}{12},\frac{7π}{12}\)

Hint:

Direction cosines squared add to 1.

Answer 1

The Correct Option is A

Step 1: Component form:
The unit vector is $\left(\frac1{\sqrt2},\ \frac12,\ \cos\theta\right)$ with magnitude 1.

Step 2: Square and add:
$\frac12+\frac14+\cos^2\theta=1$, so $\cos^2\theta=\frac14$.

Step 3: Angles:
$\cos\theta=\frac12$ gives $\frac\pi3$ and $\cos\theta=-\frac12$ gives $\frac{2\pi}3$. Option (A).

Final Answer:
cos^2 theta = 1/4. \[ \boxed{A} \]