Question

If \( p = \hat{i} + \hat{j}, q = 4\hat{k} - \hat{j} \) and \( r = \hat{i} + \hat{k} \), then the unit vector in the direction of \( 3p + q - 2r \) is

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

Hint:

To find the unit vector in a given direction, first calculate the vector, then divide by its magnitude.

Answer 1

The Correct Option is A

We are given the vectors: \[ p = \hat{i} + \hat{j}, \quad q = 4\hat{k} - \hat{j}, \quad r = \hat{i} + \hat{k} \] We need to find the unit vector in the direction of \( 3p + q - 2r \). First, calculate \( 3p + q - 2r \): \[ 3p = 3(\hat{i} + \hat{j}) = 3\hat{i} + 3\hat{j} \] \[ q = 4\hat{k} - \hat{j} \] \[ -2r = -2(\hat{i} + \hat{k}) = -2\hat{i} - 2\hat{k} \] Now, combine the vectors: \[ 3p + q - 2r = (3\hat{i} + 3\hat{j}) + (4\hat{k} - \hat{j}) + (-2\hat{i} - 2\hat{k}) \] \[ = (3\hat{i} - 2\hat{i}) + (3\hat{j} - \hat{j}) + (4\hat{k} - 2\hat{k}) \] \[ = \hat{i} + 2\hat{j} + 2\hat{k} \] Now, the magnitude of the vector is: \[ |\hat{i} + 2\hat{j} + 2\hat{k}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3 \] Thus, the unit vector is: \[ \frac{1}{3} (\hat{i} + 2\hat{j} + 2\hat{k}) \]