Let a = i +2 j + k and b =2 i + j - k . Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a . Then such a vector c is :

Correct answer: D. 1 2 (- i + k ) Let vector p in plane of a \& b =K( a +λ b ) aligned & p a = p · a =0 & ~K( a +λ b ) · a =0 & a · a +λ b · a =0 & 6+λ(3)=0 & λ=-2 & p =(-3 i +3 k ) aligned Unit vector arrow ± (- i + k ) 2

JEE Main 20258 AprilEvening Shift

Exam: JEE Main
Year: 2025
Date: 8 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2025 · 8 April · Evening Shift · Mathematics · Vector Algebra

Let a = i +2 j + k and b =2 i + j - k . Let c be a unit vector in...

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 :

Options

A $\frac{1}{\sqrt{5}}(\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$
B $\frac{1}{\sqrt{3}}(-\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})$
C $\frac{1}{\sqrt{3}}(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$
D $\frac{1}{\sqrt{2}}(-\hat{\mathrm{i}}+\hat{\mathrm{k}})$

Correct answer

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

Step-by-step solution

Let vector

\vec{p}

in plane of

\vec{a} \& \vec{b}=K(\vec{a}+\lambda \vec{b})

\begin{aligned}<br/>& \overrightarrow{\mathrm{p}} \perp \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{a}}=0 \\ & \Rightarrow \mathrm{~K}(\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{a}}=0 \\ & \Rightarrow \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}+\lambda \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}}=0 \\ & \Rightarrow 6+\lambda(3)=0 \\ & \Rightarrow \lambda=-2 \\ & \Rightarrow \overrightarrow{\mathrm{p}}=(-3 \hat{\mathrm{i}}+3 \hat{\mathrm{k}})<br/>\end{aligned}

Unit vector

\rightarrow \pm \frac{(-\hat{\mathrm{i}}+\hat{\mathrm{k}})}{\sqrt{2}}

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