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: Write as a single sine:
$\sin\theta - \sqrt3\cos\theta = 2\sin\left(\theta - \frac\pi3\right)$.

Step 2: Set to zero:
$\sin(\theta - \frac\pi3) = 0$ gives $\theta - \frac\pi3 = n\pi$. In $(0,\frac\pi2)$ the only solution is $n=0$, so $\theta = \frac\pi3$.

Step 3: Check the other options:
At $\frac\pi6$ the expression is $\frac12 - \frac32 = -1$. At $\frac\pi4$ it is $\frac{1-\sqrt3}{\sqrt2}$, nonzero. At $\frac\pi2$ it is $1$. Only $\frac\pi3$ gives zero.

Final Answer:
Option (A). \[ \boxed{\theta=\frac{\pi}{3} \text{ (A)}} \]