Question

Find the perpendicular unit vectors on the vectors \[ \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \quad \text{and} \quad \mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k} \] and find the sine of the angle between them.

Hint:

The sine of the angle between two vectors is given by \( \sin \theta = \frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}| |\mathbf{b}|} \).

Answer 1

Step 1: Compute the cross product \( \mathbf{a} \times \mathbf{b} \). \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
2 & -1 & 1
3 & 4 & -1 \end{vmatrix} \] \[ = \hat{i}((-1)(-1) - (1)(4)) - \hat{j}((2)(-1) - (1)(3)) + \hat{k}((2)(4) - (-1)(3)) \] \[ = \hat{i}(1 - 4) - \hat{j}(-2 - 3) + \hat{k}(8 + 3) \] \[ = -3\hat{i} + 5\hat{j} + 11\hat{k} \] Step 2: Compute unit vector. \[ |\mathbf{a} \times \mathbf{b}| = \sqrt{(-3)^2 + (5)^2 + (11)^2} = \sqrt{9 + 25 + 121} = \sqrt{155} \] \[ \text{Unit vector} = \frac{-3\hat{i} + 5\hat{j} + 11\hat{k}}{\sqrt{155}} \] Step 3: Compute sine of angle. \[ \sin \theta = \frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}| |\mathbf{b}|} \] \[ \sin \theta = \frac{\sqrt{155}}{\sqrt{6} \cdot \sqrt{26}} = \frac{\sqrt{155}}{\sqrt{156}} \]