Question

If \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \hat{i} + \hat{j} - \hat{k} \), then \( \vec{a} \) and \( \vec{b} \) are:

• A. collinear vectors which are not parallel
• B. parallel vectors
• C. perpendicular vectors
• D. unit vectors

Hint:

Two vectors are perpendicular if their dot product equals zero.

Answer 1

The Correct Option is C

Step 1: Find the dot product of \( \vec{a} \) and \( \vec{b} \)
The dot product of \( \vec{a} \) and \( \vec{b} \) is: \[ \vec{a} \cdot \vec{b} = (2)(1) + (-1)(1) + (1)(-1) = 2 - 1 - 1 = 0. \] 
Step 2: Check for perpendicularity
If \( \vec{a} \cdot \vec{b} = 0 \), the vectors are perpendicular. 
Step 3: Conclude the result
The vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular.