Question

If|a*b|=a•b, what is the angle between the two vectors

Answer 1

$\huge{\star}{\underline{\boxed{\red{\sf{Answer :}}}}}{\star}$

Given :-

|a × b| = |a.b|

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To find :-

Angle between two vectors.

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Solution :-

As we know that,

$\Large \implies {\boxed{\sf{a \: \times \: b \: = \: ab \: Sin \theta}}}$

And,

$\Large \implies{\boxed {\sf{a \: . \: b \: = \: ab \: Cos \theta}}}$

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Now,

$\Large \leadsto {\sf{\cancel{ab} \: Sin \theta \: = \: \cancel{ab} \: Cos \theta}}$

$\Large \leadsto {\sf{ \cancel{Cos} (90^{\circ} \: - \: \theta) \: = \: \cancel{Cos} \: \theta }}$

$\Large \leadsto {\sf{90^{\circ} \: - \: \theta \: = \: \theta}}$

$\Large \leadsto {\sf{90^{\circ} \: = \: \theta \: + \: \theta }}$

$\Large \leadsto {\sf{90^{\circ} \: = \: 2 \theta}}$

$\Large \leadsto {\sf{\theta \: = \: \frac{\cancel{90}}{\cancel{2}}}}$

$\Huge \implies {\boxed{\boxed{\sf{ \theta \: = \: 45^{\circ}}}}}$

∴ Angle between them is 45°

Answer 2

$\Huge{\underline{\underline{\mathfrak{Answer \colon}}}}$

Given

$\large{ \sf{ |a \times b| = a.b }}$

Dot Product is the scalar representation of two vectors :

$\huge{\sf{a.b = ab cos \theta}}$

Cross Product of two vectors gives a vector :

$\sf{ \vec{a} \times \vec{b} = | \vec{a}|. | \vec{b}| sin \theta } \\ \\ \huge{\leadsto \: \sf{ |a \times b| = ab \: sin \theta}}$

Now,

$\sf{ \cancel{ab} \: cos \theta \: = \cancel{ab} \: sin \theta} \\ \\ \implies \: \sf{ \frac{sin \theta}{ cos \theta} = 1 } \\ \\ \implies \: \sf{tan \theta \: = 1} \\ \\ \implies \: \sf{tan \theta \: = tan45} \\ \\ \implies \huge{\boxed{\boxed{ \: \mathtt{ \theta = 45 \degree}}}} \:$

Thus,the angle between the two vectors is 45°