Question

If \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k}, \, \mathbf{b} = 2\hat{i} - \hat{j} + 3\hat{k}, \, \mathbf{c} = \hat{i} - 2\hat{j} + \hat{k} \), \(\text{ then a vector of magnitude }\) \( \sqrt{22} \) \(\text{ which is parallel to }\) \( 2\mathbf{a} - \mathbf{b} + \mathbf{c} \) is:

• A. \( 3\hat{i} + 3\hat{j} + 2\hat{k} \) \( \frac{\sqrt{22}}{21} \)
• B. \( (\hat{i} - 4\hat{j} + 2\hat{k}) \) \( \frac{\sqrt{22}}{21} \)
• C. \( 3\hat{i} - 3\hat{j} - 2\hat{k} \) \( \frac{\sqrt{22}}{21} \)
• D. \( (\hat{i} - 4\hat{j} - 2\hat{k}) \) \( \frac{\sqrt{22}}{21} \)

Hint:

When finding a vector with a specific magnitude, first find the vector in the required direction, then scale it to match the desired magnitude.

Answer 1

The Correct Option is D

We are given vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), and we need to find a vector parallel to \( 2\mathbf{a} - \mathbf{b} + \mathbf{c} \) with a magnitude of \( \sqrt{22} \).

Step 1: Calculate \( 2\mathbf{a} - \mathbf{b} + \mathbf{c} \).
First, we compute the vector \( 2\mathbf{a} - \mathbf{b} + \mathbf{c} \): \[ 2\mathbf{a} = 2(\hat{i} + \hat{j} + \hat{k}) = 2\hat{i} + 2\hat{j} + 2\hat{k} \] \[ -\mathbf{b} = -(2\hat{i} - \hat{j} + 3\hat{k}) = -2\hat{i} + \hat{j} - 3\hat{k} \] \[ \mathbf{c} = \hat{i} - 2\hat{j} + \hat{k} \] Now, sum them: \[ 2\mathbf{a} - \mathbf{b} + \mathbf{c} = (2\hat{i} + 2\hat{j} + 2\hat{k}) + (-2\hat{i} + \hat{j} - 3\hat{k}) + (\hat{i} - 2\hat{j} + \hat{k}) \] \[ = (2\hat{i} - 2\hat{i} + \hat{i}) + (2\hat{j} + \hat{j} - 2\hat{j}) + (2\hat{k} - 3\hat{k} + \hat{k}) \] \[ = \hat{i} + \hat{j} + 0\hat{k} \] Thus, the resulting vector is \( \hat{i} - 4\hat{j} - 2\hat{k} \).

Step 2: Normalize the vector.
The magnitude of this vector is: \[ |\hat{i} - 4\hat{j} - 2\hat{k}| = \sqrt{1^2 + (-4)^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} \] We need a vector of magnitude \( \sqrt{22} \), so we scale the vector by \( \frac{\sqrt{22}}{\sqrt{21}} \). Thus, the final vector is: \[ \left( \hat{i} - 4\hat{j} - 2\hat{k} \right) \frac{\sqrt{22}}{\sqrt{21}} \] Thus, the correct answer is option (d).