Question

If |

⃗A⃗ × B⃗ | = |

⃗A⃗ ∙ B⃗ | then angle between ⃗A⃗ and B⃗ is ……​

Answer 1

SOLUTION

GIVEN

$\sf{ |\vec{A} \times \vec{B}| = |\vec{A} . \vec{B}| }$

TO DETERMINE

The angle between the vectors

$\sf{ \vec{A} \: \: \: and \: \: \: \vec{B} \: }$

EVALUATION

Let θ be the angle between the vectors

$\sf{ \vec{A} \: \: \: and \: \: \: \vec{B} \: }$

Then

$\sf{ |\vec{A} \times \vec{B}| = |\vec{A} || \vec{B}| \sin \theta }$

$\sf{ |\vec{A} . \vec{B}| = |\vec{A} || \vec{B}| \cos \theta }$

So by the given condition

$\sf{ |\vec{A} \times \vec{B}| = |\vec{A} . \vec{B}| }$

$\sf{ \implies |\vec{A} || \vec{B}| \sin \theta = |\vec{A} || \vec{B}| \cos \theta }$

$\sf{ \implies \sin \theta =\cos \theta }$

$\sf{ \implies \tan \theta =1}$

$\displaystyle \sf{ \implies \theta = \frac{\pi}{4} }$

FINAL ANSWER

Hence the required angle is

$\displaystyle \sf{ \frac{\pi}{4} }$

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