Let vec a and vec b be two unit vectors such that the angle between them is π/3 . If λ vec a +2 vec b and 3 vec a -λ vec b are perpendicular to each other, then the number of values of λ in [-1,3] is :

Mathematics · Vector Algebra

JEE Main 2025 — 22 January, Evening Shift — Question 15

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$ . If $\lambda \vec{a}+2 \vec{b}$ and $3 \vec{a}-\lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is :

Option A:
3
Option B:
2
Option C:
1
Option D:
0
Correct

Answer: D

Step-by-step solution

$\quad \hat{a} . \hat{b}=\frac{1}{2}$

Now $(\lambda \hat{a}+2 \hat{b}) \cdot(3 \hat{a}-\lambda \hat{b})=0$

$3 \lambda \hat{a} . \hat{a}-\lambda^{2} \hat{a} . \hat{b}+6 \hat{a} . \hat{b}-2 \lambda \hat{b} . \hat{b}=0$

$3 \lambda-\frac{\lambda^{2}}{2}+3-2 \lambda=0$

$\lambda^{2}-2 \lambda-6=0$

$\lambda=1 \pm \sqrt{7}$

$\Rightarrow$ number of values $=0$

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 22 January, Evening Shift
Subject: Mathematics
Chapter: Vector Algebra
Topic: Scalar or Dot Product of Two Vectors

Let a line pass through two distinct points P(-2,-1,3) and Q , and be parallel to the vector 3 hati+2 hatj+2 k . If the distance of the…

Answer: D

Step-by-step solution

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