Question

The number of unit vectors perpendicular to \(\vec{a} = \hat{i}+\hat{j}\) and \(\vec{b} = \hat{j}+\hat{k}\) is:

• A. infinite
• B. one
• C. two
• D. three

Hint:

The cross product gives a direction perpendicular to both vectors. Normalization yields two unit vectors (opposite directions).

Answer 1

The Correct Option is C

Concept: A vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) is parallel to \(\vec{a} \times \vec{b}\).

Step 1: Compute \(\vec{a} \times \vec{b}\). \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} 1 & 1 & 0 0 & 1 & 1 \end{vmatrix} = \hat{i}(1\cdot1 - 0\cdot1) - \hat{j}(1\cdot1 - 0\cdot0) + \hat{k}(1\cdot1 - 1\cdot0) \] \[ = \hat{i}(1) - \hat{j}(1) + \hat{k}(1) = \hat{i} - \hat{j} + \hat{k} \]

Step 2: Unit vectors. \[ |\vec{a} \times \vec{b}| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3} \] The two unit vectors are: \[ \pm \frac{\hat{i} - \hat{j} + \hat{k}}{\sqrt{3}} \]