Question

The unit vector perpendicular to the vectors \( \hat{i} - \hat{j} \) and \( \hat{i} + \hat{j} \) is:

Hint:

The cross product of two vectors gives a vector perpendicular to both. To obtain a unit vector, divide the cross product by its magnitude.

Answer 1

Let \( \vec{a} = \hat{i} - \hat{j} \) and \( \vec{b} = \hat{i} + \hat{j} \). The unit vector perpendicular to both vectors is given by the cross product \( \vec{a} \times \vec{b} \). We compute the cross product: \[ \vec{a} \times \vec{b} = \left( \hat{i} - \hat{j} \right) \times \left( \hat{i} + \hat{j} \right) \] Using the distributive property and properties of unit vectors: \[ \vec{a} \times \vec{b} = \hat{i} \times \hat{i} + \hat{i} \times \hat{j} - \hat{j} \times \hat{i} - \hat{j} \times \hat{j} \] Since \( \hat{i} \times \hat{i} = 0 \), \( \hat{j} \times \hat{j} = 0 \), and \( \hat{i} \times \hat{j} = \hat{k} \), we get: \[ \vec{a} \times \vec{b} = \hat{k} + \hat{k} = 2 \hat{k} \] Thus, the unit vector perpendicular to both vectors is: \[ \frac{2 \hat{k}}{|2 \hat{k}|} = \hat{k} \] Therefore, the correct answer is \( \hat{k} \).