Let the angle θ, 0<θ<π/2 between two unit vectors hat a and hat b be sin -1 (frac √(65) 9 ) . If the vector vec c =3 hat a +6 hat b +9(hat a × hat b ), then the value of 9(vec c . hat a )-3(vec c . hat b ) is

Mathematics · Vector Algebra

JEE Main 2025 — 7 April, Morning Shift — Question 30

Let the angle $\theta, 0<\theta<\frac{\pi}{2}$ between two unit vectors $\hat{a}$ and $\hat{b}$ be $\sin ^{-1}\left(\frac{\sqrt{65}}{9}\right)$ . If the vector $\vec{c}=3 \hat{a}+6 \hat{b}+9(\hat{a} \times \hat{b}), \quad$ then the value of $9(\vec{c} . \hat{a})-3(\vec{c} . \hat{b})$ is

Option A:
24
Option B:
27
Option C:
31
Option D:
29
Correct

Answer: D

Step-by-step solution

$\vec{c} . \hat{a}=3+6 \hat{a} . \hat{b}$

$\vec{c} \cdot \hat{b}=3 \hat{a} \cdot \hat{b}+6$

So, $9(\vec{c} . \hat{a})-3(\vec{c} . \hat{b})=27-18+(54-9) \hat{a} . \hat{b}$

\begin{aligned} & =9+45 \hat{a} . \hat{b} \\& =9+45 \times \frac{\sqrt{81-65}}{9} \\& =9+5 \sqrt{16}=29 \end{aligned}

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 7 April, Morning Shift
Subject: Mathematics
Chapter: Vector Algebra
Topic: Scalar or Dot Product of Two Vectors