Question

If the position vectors of the points \( A, B, C, D \) are successively \( \hat{i} + \hat{j} + \hat{k} \), \( 2\hat{i} + 5\hat{j} \), \( 3\hat{i} + 2\hat{j} - 3\hat{k} \) and \( \hat{i} - 6\hat{j} + \hat{k} \), then find the angle between the lines \( AB \) and \( CD \).

Hint:

To find the angle between two vectors, use \( \cos\theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \).

Answer 1

To find the angle between two lines, we first determine the direction vectors: \[ \overrightarrow{AB} = (2\hat{i} + 5\hat{j}) - (\hat{i} + \hat{j} + \hat{k}) = \hat{i} + 4\hat{j} - \hat{k}. \] \[ \overrightarrow{CD} = (\hat{i} - 6\hat{j} + \hat{k}) - (3\hat{i} + 2\hat{j} - 3\hat{k}) = -2\hat{i} - 8\hat{j} + 4\hat{k}. \] The angle \( \theta \) is given by: \[ \cos\theta = \frac{\overrightarrow{AB} \cdot \overrightarrow{CD}}{|\overrightarrow{AB}| |\overrightarrow{CD}|}. \] Computing the dot product and magnitudes, we find: \[ \theta = \cos^{-1} \left( \frac{-10}{\sqrt{18} \cdot \sqrt{84}} \right). \]