A and B are expressed as A =2 i + j and B = i - j . Unit vector perpendicular to A and B is

Correct answer: D. - k Given vectors are A = 2 i + j and B = i - j . A vector perpendicular to both A and B is given by their cross product A × B . A × B = (2 i + j ) × ( i - j ) A × B = 2( i × i ) - 2( i × j ) + ( j × i ) - ( j × j ) Since i × i = 0 , j × j = 0 , i × j = k , and j × i = - k : A × B = 0 - 2 k - k - 0 = -3 k The magnitude of this vector is | A × B | = |-3 k | = 3 . The unit vector perpendicular to both A and B is: n = A × B | A × B | = -3 k 3 = - k Answer: - k

MHT CET 202624 AprilMorning Shift

Exam: MHT CET
Year: 2026
Date: 24 April
Shift: Morning Shift
Subject: Physics
Chapter: Mathematics in Physics

MHT CET 2026 · 24 April · Morning Shift · Physics · Mathematics in Physics

A and B are expressed as A =2 i + j and B = i - j . Unit vector...

\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}

Options

A $\dfrac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$
B $\dfrac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$
C $\dfrac{\hat{i}+\hat{k}}{\sqrt{3}}$
D $-\hat{k}$

Correct answer

D. $-\hat{k}$

Step-by-step 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}

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