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  • Let   and <img alt="\hat{b}" src="https://learn.careers360.com/latex-image/?%5Cdpi%7B100%7

Let   \hat{a} and \hat{b} be two unit vectors. If the vectors \vec{c}= \hat{a}+2\hat{b} and \vec{d}= 5\hat{a}-4\hat{b}  are perpendicular to each other, then the angle between  \hat{a} and \hat{b} is

Option: 1

\frac{\pi }{6}


Option: 2

\frac{\pi }{2}


Option: 3

\frac{\pi }{3}


Option: 4

\frac{\pi }{4}


Answers (1)

best_answer

\vec{c}= \hat{a}+2\hat{b},\vec{d}= 5\hat{a}-4\hat{b}

\therefore \vec{c}.\vec{d}= 0\Rightarrow \left ( \hat{a} +2 \hat{b}\right )\cdot \left ( 5\hat{a} -4 \hat{b}\right )=5-4\hat{b}\cdot \hat{a}+10\hat{b}\cdot \hat{a}-8

\Rightarrow 6\hat{b}\cdot \hat{a}-3= 0\Rightarrow \hat{b}\cdot \hat{a}=\frac{1}{2}\: \: \: \: \therefore \Theta = \frac{\pi }{3}

As learnt in concept

Scalar Product of two vectors (dot product) -

\vec{a}\vec{b}=\left | a \right |\left | b \right |Cos\theta

- wherein

\Theta is the angle between the vectors\vec{a}\: and\:\vec{b}

 

 

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