NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

If $\vec{A} = 4\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{B} = 2\hat{i} - \hat{j} + 2\hat{k}$ , then the unit vector $\hat{N}$ perpendicular to the vectors $\vec{A}$ and $\vec{B}$ , such that $\vec{A}, \vec{B}$ and $\hat{N}$ form a right handed system, is:

Choose the correct answer:

Explanation

Solution

Concept:

The cross product $\vec{A} \times \vec{B}$ gives a vector perpendicular to both.
The unit vector $\hat{u}$ in the direction of $\vec{A}$ is $\frac{\vec{A}}{|\vec{A}|}$ .

Calculation:

$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 3 & 1 \\ 2 & -1 & 2 \end{vmatrix}$

Expanding along $R_1$ :

$= (6+1)\hat{i} + (2-8)\hat{j} + (-4-6)\hat{k} = 7\hat{i} - 6\hat{j} - 10\hat{k}$

Magnitude $|\vec{A} \times \vec{B}| = \sqrt{7^2 + (-6)^2 + (-10)^2} = \sqrt{185}$

Unit vector $\hat{N} = \frac{\vec{A} \times \vec{B}}{|\vec{A} \times \vec{B}|} = \frac{1}{\sqrt{185}}(7\hat{i} - 6\hat{j} - 10\hat{k})$

Correct Option: 1

Choose the correct answer:

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