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UPSC NDA Math Model Paper 1

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Question : 17 of 120

Marks: +1, -0

If

\left|\overset{\rightarrow}{a}\times \overset{\rightarrow}{b}\right|=\left|\overset{\rightarrow}{a}\cdot \overset{\rightarrow}{b}\right|

,then angle between

\overset{\rightarrow}{a}

\overset{\rightarrow}{b}

is

90°
60°
45°
30°

Solution:

Concept:

Let

\overset{\rightarrow}{a}

and

\overset{\rightarrow}{b}

be the two vectors,

Dot product of two vectors is given by:

\overset{\rightarrow}{a}\cdot \overset{\rightarrow}{b}=\left|\overset{\rightarrow}{a}\right|\left|\overset{\rightarrow}{b}\right| \cos \theta

Cross product of two vectors is given by:

\overset{\rightarrow}{a}\times\overset{\rightarrow}{b}=\left|\overset{\rightarrow}{a}\right|\left|\overset{\rightarrow}{b}\right|\sin\theta \hat{n}

, where

\hat{n}

is a unit vector

Calculation:

Given:

\left|\overset{\rightarrow}{a}\times \overset{\rightarrow}{b}\right|=\left|\overset{\rightarrow}{a}\right|\left|\overset{\rightarrow}{b}\right|\cos\theta

To Find: Angle between

\overset{\rightarrow}{a}

adn

\overset{\rightarrow}{b}

\left|\overset{\rightarrow}{a}\times \overset{\rightarrow}{b}\right|=\left|\overset{\rightarrow}{a}\cdot \overset{\rightarrow}{b}\right|

\Rightarrow \left|\overset{\rightarrow}{a}\right|\left|\overset{\rightarrow}{b}\right| \sin \theta |\hat{n}|=\left|\overset{\rightarrow}{a}\right|\left|\overset{\rightarrow}{b}\right| \cos \theta

\Rightarrow \sin \theta=\cos \theta \;\; (\because |\hat{n}|=1)

\Rightarrow \tan \theta=1

\therefore \theta=45^{\circ}

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