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Vector Algebra question

2025 · 7 Apr · Shift 1 · Q44

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JEE Main
Mathematics
Vector Algebra
2025 · 7 Apr · Shift 1 · Q44

Vector Algebra question

2025 · 7 Apr · Shift 1 · Q44

JEE MainMathematicsVector AlgebraMCQ+4 / −1

Let the angle

\theta, 0\lt \theta\lt \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})

, then the value of

9(\vec{c} \cdot \hat{a})-3(\vec{c} \cdot \hat{b})

is

A
31
B
29
C
24
D
27

View written solutionFree

Correct answer: D

Given data

$\hat a$ and $\hat b$ are unit vectors.
Angle between them is $\theta=\sin^{-1}\left(\frac{\sqrt{65}}{9}\right), \qquad 0<\theta<\frac{\pi}{2}$
$\vec c=3\hat a+6\hat b+9(\hat a\times \hat b)$

We need to find: $9(\vec c\cdot \hat a)-3(\vec c\cdot \hat b)$

Find $\cos\theta$

Since $\sin\theta=\frac{\sqrt{65}}{9}$ we get

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