Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is

(D) $\frac{1}{\sqrt{2}}(-\hat{i}+\hat{k})$

(A) $\frac{1}{\sqrt{5}}(\hat{j}-2 \hat{k})$

(B) $\frac{1}{\sqrt{3}}(-\hat{i}+\hat{j}-\hat{k})$

(C) $\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$

JEE Main · 2025 · 08 Apr 2025 · Shift 2 · Vector Algebra

Let

a = \hat{i} + 2 \hat{j} + \hat{k}

and

b = 2 \hat{i} + \hat{j} - \hat{k}

. Let

\overset{c}{^}

be a unit vector in the plane of the vectors

a

and

b

and be perpendicular to

a

. Then such a vector

\overset{c}{^}

Answer: (D)

A vector coplanar with

a

and

b

has the form

a + λ b

. Perpendicularity to

a

requires

(a + λ b) \cdot a = 0

; with

a \cdot a = 6

and

b \cdot a = 3

, this gives

6 + 3 λ = 0

, so

λ = - 2

. Then

a - 2 b = (1 - 4) \hat{i} + (2 - 2) \hat{j} + (1 + 2) \hat{k} = - 3 \hat{i} + 3 \hat{k}

. Normalising,

\overset{c}{^} = \frac{- 3 i ^ + 3 k ^}{3 2} = \frac{1}{2} (- \hat{i} + \hat{k})

JEE Main · 2025 · 08 Apr 2025 · Shift 2 · Single correct option