Question

If θ is the angle between any two vectors \(\vec a\) and \(\vec b\) , then|\(\vec a.\vec b\)|=|\(\vec a \times \vec b\)| when θ is equal to

Answer 1

Let θ be the angle between two vectors \(\vec a\) and \(\vec b\).
Then, without loss of generality,\(\vec a\)and \(\vec b\)are non-zero vectors,so
that |\(\vec a\)|and |\(\vec b\)|are positive.
|\(\vec a.\vec b\)|=|\(\vec a \times \vec b\)|
⇒ |\(\vec a\)||\(\vec b\)|cosθ=\(\vec a\)||\(\vec b\)|sinθ
⇒ cosθ=sinθ [|\(\vec a\)|and|\(\vec b\)|are positive]
⇒ tanθ=1
⇒ θ=\(\frac{\pi}{4}\)
Hence,|\(\vec a.\vec b\)|=|\(\vec a \times \vec b\)|when θ=\(\frac{\pi}{4}\)
The correct answer is B.