Question

Let $ \mathbf{a} = 2\hat{i} - 3\hat{j} + \hat{k}, \, \mathbf{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} $ and a vector $ \mathbf{c} $ be such that $ (\mathbf{a} - \mathbf{c}) \times \mathbf{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} $ and $ \mathbf{a} \cdot \mathbf{c} = 3 $. If $ \mathbf{b} \times \mathbf{c} = \mathbf{a} $, then $ |\mathbf{a} \cdot \mathbf{c}| $ is equal to:

• A. 18
• B. 12
• C. 9
• D. 15

Hint:

When working with cross and dot products, make sure to calculate each component carefully and use the appropriate properties of these operations to find relationships between vectors.

Answer 1

The Correct Option is D

Given: \[ \mathbf{a} = 2\hat{i} - 3\hat{j} + \hat{k}, \quad \mathbf{b} = 3\hat{i} + 2\hat{j} + 5\hat{k} \] The cross product \( \mathbf{a} \times \mathbf{b} \) is calculated as follows: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} 2 & -3 & 1 3 & 2 & 5 \end{vmatrix} = \hat{i} \left( (-3)(5) - (1)(2) \right) - \hat{j} \left( (2)(5) - (1)(3) \right) + \hat{k} \left( (2)(2) - (-3)(3) \right) \] \[ = \hat{i}(-15 - 2) - \hat{j}(10 - 3) + \hat{k}(4 + 9) \] \[ = -17\hat{i} - 7\hat{j} + 13\hat{k} \] So, \[ \mathbf{a} \times \mathbf{b} = -17\hat{i} - 7\hat{j} + 13\hat{k} \] We are given that: \[ (\mathbf{a} - \mathbf{c}) \times \mathbf{b} = -18\hat{i} - 3\hat{j} + 12\hat{k} \] Thus, we have: \[ (\mathbf{a} - \mathbf{c}) \times \mathbf{b} = \mathbf{a} \times \mathbf{b} - \mathbf{c} \times \mathbf{b} \] \[ -18\hat{i} - 3\hat{j} + 12\hat{k} = -17\hat{i} - 7\hat{j} + 13\hat{k} - \mathbf{c} \times \mathbf{b} \] So, \[ \mathbf{c} \times \mathbf{b} = -\hat{i} + 4\hat{j} - \hat{k} \] Now, we use the condition \( \mathbf{b} \times \mathbf{c} = \mathbf{a} \): \[ \mathbf{b} \times \mathbf{c} = (3\hat{i} + 2\hat{j} + 5\hat{k}) \times \mathbf{c} = 2\hat{i} - 3\hat{j} + \hat{k} \] Thus, \[ \mathbf{c} \times \mathbf{b} = \mathbf{a} \implies \mathbf{c} = \left( \mathbf{a} \cdot \mathbf{b} \right) \] Finally, we calculate: \[ \mathbf{a} \cdot \mathbf{c} = -2 - 12 - 1 = -15 \] Hence, \( |\mathbf{a} \cdot \mathbf{c}| = 15 \).