Question

\(\vec{A}\) and \(\vec{B}\) are expressed as \(\vec{A}=2\hat{i}+\hat{j}\) and \(\vec{B}=\hat{i}-\hat{j}\). Unit vector perpendicular to \(\vec{A}\) and \(\vec{B}\) is

• A. \(\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}\)
• B. \(\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}\)
• C. \(\frac{\hat{i}+\hat{k}}{\sqrt{3}}\)
• D. \(-\hat{k}\)

Hint:

A vector perpendicular to two vectors is along their cross product. Divide it by its magnitude to get the unit vector.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A vector perpendicular to both \(\vec{A}\) and \(\vec{B}\) is along their cross product \(\vec{A}\times\vec{B}\).

Step 2: Calculation
\(\vec{A}\times\vec{B}=(2\hat{i}+\hat{j})\times(\hat{i}-\hat{j})\)
\(=-2\hat{k}-\hat{k}=-3\hat{k}\)
Its magnitude is \(3\).

Step 3: Unit vector
Divide by the magnitude: \(\dfrac{-3\hat{k}}{3}=-\hat{k}\).

Final Answer:
The unit vector perpendicular to both vectors is \(-\hat{k}\). \[ \boxed{\text{(D) }-\hat{k}} \]