Question

Let $ \vec{a} = \hat{i} + \hat{j} + \hat{k} $, $ \vec{b} = 3\hat{i} + 2\hat{j} - \hat{k} $, $ \vec{c} = \lambda \hat{j} + \mu \hat{k} $ and $ \hat{d} $ be a unit vector such that $ \vec{a} \times \hat{d} = \vec{b} \times \hat{d} $ and $ \vec{c} \cdot \hat{d} = 1 $. If $ \vec{c} $ is perpendicular to $ \vec{a} $, then $ |3\lambda \hat{d} + \mu \vec{c}|^2 $ is equal to ____.

Hint:

Use the given vector relations to find the vectors \( \hat{d} \) and \( \vec{c} \). Remember that if \( \vec{a} \times \vec{b} = 0 \), then \( \vec{a} \) and \( \vec{b} \) are parallel.

Answer 1

Correct Answer: 5

From \( \vec{a} \times \hat{d} = \vec{b} \times \hat{d} \), we deduce \( (\vec{a} - \vec{b}) \times \hat{d} = 0 \), implying \( \vec{a} - \vec{b} \) is parallel to \( \hat{d} \). Calculating \( \vec{a} - \vec{b} \): \[ \vec{a} - \vec{b} = (\hat{i} + \hat{j} + \hat{k}) - (3\hat{i} + 2\hat{j} - \hat{k}) = -2\hat{i} - \hat{j} + 2\hat{k} \] Thus, \( \hat{d} \) is determined as: \[ \hat{d} = \frac{\vec{a} - \vec{b}}{|\vec{a} - \vec{b}|} = \frac{-2\hat{i} - \hat{j} + 2\hat{k}}{\sqrt{(-2)^2 + (-1)^2 + 2^2}} = \frac{-2\hat{i} - \hat{j} + 2\hat{k}}{3} \] Given \( \vec{c} \cdot \hat{d} = 1 \), and substituting \( \vec{c} = \lambda \hat{j} + \mu \hat{k} \): \[ (\lambda \hat{j} + \mu \hat{k}) \cdot \frac{-2\hat{i} - \hat{j} + 2\hat{k}}{3} = 1 \] \[ \frac{-\lambda + 2\mu}{3} = 1 \Rightarrow -\lambda + 2\mu = 3 \quad (1) \] Given \( \vec{c} \) is perpendicular to \( \vec{a} \), so \( \vec{c} \cdot \vec{a} = 0 \): \[ (\lambda \hat{j} + \mu \hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) = 0 \] \[ \lambda + \mu = 0 \Rightarrow \mu = -\lambda \quad (2) \] Substituting (2) into (1): \[ -\lambda - 2\lambda = 3 \Rightarrow -3\lambda = 3 \Rightarrow \lambda = -1, \quad \mu = 1 \] Therefore, \( \vec{c} = -\hat{j} + \hat{k} \). Now, we compute \( 3\lambda \hat{d} + \mu \vec{c} \): \[ 3\lambda \hat{d} + \mu \vec{c} = 3(-1) \cdot \frac{-2\hat{i} - \hat{j} + 2\hat{k}}{3} + (-\hat{j} + \hat{k}) \] \[ = 2\hat{i} + \hat{j} - 2\hat{k} - \hat{j} + \hat{k} = 2\hat{i} - \hat{k} \] Finally, the magnitude squared: \[ |3\lambda \hat{d} + \mu \vec{c}|^2 = |2\hat{i} - \hat{k}|^2 = 2^2 + (-1)^2 = 4 + 1 = 5 \]