Question

Which one of the following is a unit vector?

• A. \( \cos \theta \hat{i} + \sin \theta \hat{j} \)
• B. \( \frac{1}{\sqrt{3}} (\hat{i} + \hat{j}) \)
• C. \( 2 \hat{i} - 3 \hat{j} \)
• D. \( \sin \theta \hat{i} - 2 \cos \theta \hat{j} \)

Hint:

To check if a vector is a unit vector, compute its magnitude and ensure it is equal to 1.

Answer 1

The Correct Option is A

Step 1: Definition of a Unit Vector.
A unit vector is a vector that has a magnitude of 1. To verify if a given vector is a unit vector, we compute its magnitude and check if it equals 1. The magnitude of the vector \( \cos \theta \hat{i} + \sin \theta \hat{j} \) is: \[ \sqrt{(\cos^2 \theta + \sin^2 \theta)} = \sqrt{1} = 1 \] Thus, the vector is a unit vector. Step 2: Final Answer.
Thus, the unit vector is \( \cos \theta \hat{i} + \sin \theta \hat{j} \).