Question

The value of \(θ\in (0,\frac{π}{2})\) for which vectors \(\overset{̄}{a} = (sinθ)\hat{i}+(cosθ)\hat{j}\) and \(\overset{̄}{b} = \hat{i}-\sqrt{3}\hat{j}+2\hat{k}\) are perpendicular is

• A. \(θ = \frac{π}{3}\)
• B. \(θ = \frac{π}{6}\)
• C. \(θ = \frac{π}{4}\)
• D. \(θ = \frac{π}{2}\)

Hint:

Perpendicular vectors have zero dot product. Solve sin(theta) - sqrt(3) cos(theta) = 0.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Two vectors are perpendicular when their dot product is zero.

Step 2: Key Formula or Approach:
\(\bar a\cdot\bar b = (\sin\theta)(1) + (\cos\theta)(-\sqrt3) + (0)(2)\).

Step 3: Detailed Explanation:
\[ \sin\theta - \sqrt3\cos\theta = 0 \Rightarrow \tan\theta = \sqrt3 \]
For \(\theta \in \left(0, \dfrac\pi2\right)\), the angle with tangent \(\sqrt3\) is \(\theta = \dfrac\pi3\).
Check: \(\sin\frac\pi3 = \frac{\sqrt3}{2}\) and \(\sqrt3\cos\frac\pi3 = \frac{\sqrt3}{2}\), which cancel.
Option (B) \(\frac\pi6\) has \(\tan = \frac1{\sqrt3}\), option (C) \(\frac\pi4\) has \(\tan = 1\), and option (D) \(\frac\pi2\) is not in the open interval and has no tangent.

Final Answer:
\(\theta = \dfrac\pi3\), option (A). \[ \boxed{\theta=\frac{\pi}{3} \text{ (A)}} \]