Two unit vectors ' a _ 1 'and ' a _ 2 ' are inclined to each other at an angle ' θ '. If | a _ 1 - a _ 2 |= 3 , then the value of ( a _ 1 - a _ 2 ) ·(2 a _ 1 - a _ 2 ) is

Correct answer: A. (1)/(2) | a _ 1 - a _ 2 |= 3 arrow(1)| a _ 1 |=| a _ 2 |=1 a _ 1 ^ 2 + a _ 2 ^ 2 -2 a _ 1 a _ 2 θ=3 squaring (1) 1+1-2 θ=3 =( a _ 1 - a _ 2 ) ·(2 a _ 1 - a _ 2 ) θ=-1 / 2

JEE Main

Exam: JEE Main
Subject: Physics
Chapter: Mathematics in Physics

JEE Main · Physics · Mathematics in Physics

Two unit vectors ' a _ 1 'and ' a _ 2 ' are inclined to each other...

Two unit vectors '

\hat{a}_{1}

'and '

\hat{a}_{2}

' are inclined to each other at an angle '

\theta

'. If \left|\hat{a}_{1}-\hat{a}_{2}\right|=\sqrt{3}

, then the value of \left(\hat{a}_{1}-\hat{a}_{2}\right) \cdot\left(2 \hat{a}_{1}-\hat{a}_{2}\right) is

Options

A $\frac{1}{2}$
B2
C1
D $\frac{3}{2}$

Correct answer

A. $\frac{1}{2}$

Step-by-step solution

\left|\hat{a}_{1}-\hat{a}_{2}\right|=\sqrt{3} \rightarrow(1)\left|\hat{a}_{1}\right|=\left|\hat{a}_{2}\right|=1\hat{a}_{1}^{2}+\hat{a}_{2}^{2}-2 \hat{a}_{1} \hat{a}_{2} \cos \theta=3 \quad

squaring (1)

1+1-2 \cos \theta=3 \quad=\left(\hat{a}_{1}-\hat{a}_{2}\right) \cdot\left(2 \hat{a}_{1}-\hat{a}_{2}\right)\cos \theta=-1 / 2

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