Q176 / 200 · MHT CET - Medical 2026 (24 April Shift 1) · Physics
$\vec{A}$ and $\vec{B}$ are expressed as $\vec{A}=2\hat{i}+\hat{j}$ and $\vec{B}=\hat{i}-\hat{j}$. Unit vector perpendicular to $\vec{A}$ and $\vec{B}$ is
- A$\dfrac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$
- B$\dfrac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$
- C$\dfrac{\hat{i}+\hat{k}}{\sqrt{3}}$
- Correct answerD$-\hat{k}$
Solution
Given vectors are $\vec{A} = 2\hat{i} + \hat{j}$ and $\vec{B} = \hat{i} - \hat{j}$.
A vector perpendicular to both $\vec{A}$ and $\vec{B}$ is given by their cross product $\vec{A} \times \vec{B}$.
$\vec{A} \times \vec{B} = (2\hat{i} + \hat{j}) \times (\hat{i} - \hat{j})$
$\vec{A} \times \vec{B} = 2(\hat{i} \times \hat{i}) - 2(\hat{i} \times \hat{j}) + (\hat{j} \times \hat{i}) - (\hat{j} \times \hat{j})$
Since $\hat{i} \times \hat{i} = 0$, $\hat{j} \times \hat{j} = 0$, $\hat{i} \times \hat{j} = \hat{k}$, and $\hat{j} \times \hat{i} = -\hat{k}$:
$\vec{A} \times \vec{B} = 0 - 2\hat{k} - \hat{k} - 0 = -3\hat{k}$
The magnitude of this vector is $|\vec{A} \times \vec{B}| = |-3\hat{k}| = 3$.
The unit vector perpendicular to both $\vec{A}$ and $\vec{B}$ is:
$\hat{n} = \dfrac{\vec{A} \times \vec{B}}{|\vec{A} \times \vec{B}|} = \dfrac{-3\hat{k}}{3} = -\hat{k}$
Answer: $-\hat{k}$
A vector perpendicular to both $\vec{A}$ and $\vec{B}$ is given by their cross product $\vec{A} \times \vec{B}$.
$\vec{A} \times \vec{B} = (2\hat{i} + \hat{j}) \times (\hat{i} - \hat{j})$
$\vec{A} \times \vec{B} = 2(\hat{i} \times \hat{i}) - 2(\hat{i} \times \hat{j}) + (\hat{j} \times \hat{i}) - (\hat{j} \times \hat{j})$
Since $\hat{i} \times \hat{i} = 0$, $\hat{j} \times \hat{j} = 0$, $\hat{i} \times \hat{j} = \hat{k}$, and $\hat{j} \times \hat{i} = -\hat{k}$:
$\vec{A} \times \vec{B} = 0 - 2\hat{k} - \hat{k} - 0 = -3\hat{k}$
The magnitude of this vector is $|\vec{A} \times \vec{B}| = |-3\hat{k}| = 3$.
The unit vector perpendicular to both $\vec{A}$ and $\vec{B}$ is:
$\hat{n} = \dfrac{\vec{A} \times \vec{B}}{|\vec{A} \times \vec{B}|} = \dfrac{-3\hat{k}}{3} = -\hat{k}$
Answer: $-\hat{k}$
Hence, the correct answer is option: 'D'.