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: Understanding the Concept:
Direction cosines of any line satisfy \(l^2+m^2+n^2=1\). For a unit vector, the direction cosines are its components.

Step 2: Substitute:
\(l=\cos\dfrac\pi4=\dfrac1{\sqrt2}\), \(m=\cos\dfrac\pi3=\dfrac12\), \(n=\cos\theta\).
\[ \frac12+\frac14+\cos^2\theta=1\ \Rightarrow\ \cos^2\theta=\frac14 \]

Step 3: Solve:
\(\cos\theta=\pm\dfrac12\). For \(\theta\in(0,\pi)\): \(\theta=\dfrac\pi3\) or \(\theta=\dfrac{2\pi}3\).

Step 4: Choose:
Option (A). Option (C) has \(\dfrac{5\pi}4\) outside \((0,\pi)\).

Final Answer:
theta is pi/3 or 2 pi/3. \[ \boxed{\frac\pi3,\ \frac{2\pi}3} \]