Question

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

• A. \( 2\hat{j} \)
• B. \( \hat{j} \)
• C. \( \frac{\hat{i} - \hat{k}}{\sqrt{2}} \)
• D. \( \frac{\hat{i} + \hat{k}}{\sqrt{2}} \)

Hint:

For finding a vector perpendicular to two given vectors, use the cross-product. To convert to a unit vector, divide by the magnitude of the resulting vector.

Answer 1

The Correct Option is B

Step 1: {Use the cross-product to find the perpendicular vector}
The cross-product of two vectors \( \vec{A} = \hat{i} + \hat{k} \) and \( \vec{B} = \hat{i} - \hat{k} \) gives a vector perpendicular to both: \[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 1 \\ 1 & 0 & -1 \\ \end{vmatrix}. \] Expanding the determinant: \[ \vec{A} \times \vec{B} = \hat{i}(0 - 0) - \hat{j}(1 - (-1)) + \hat{k}(0 - 0) = -2\hat{j}. \] Step 2: {Normalize the vector to obtain the unit vector}
The magnitude of \( -2\hat{j} \) is: \[ \|\vec{A} \times \vec{B}\| = 2. \] Thus, the unit vector is: \[ \hat{j} = \frac{-2\hat{j}}{2}. \]