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# Choose the correct answer if θ is the angle between any two vectors $\overrightarrow a$ and $\overrightarrow b$, then $| \overrightarrow a ⋅\overrightarrow b | \: = \: | \overrightarrow a \times\overrightarrow b |$ when θ is equal to

\begin{array} {1 1} (A)\; 0 \quad &(B) \;\frac{\pi}{4} \quad &(C)\; \frac{\pi}{2} \quad &(D) \;π \end{array}
Can you answer this question?

Toolbox:
• $\overrightarrow a\times \overrightarrow b=|\overrightarrow a||\overrightarrow b|\sin\theta \; \hat n$
• $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|\cos\theta$
Step 1:
Let $\theta$ be the angle between the vectors $\overrightarrow a$ and $\overrightarrow b$
Therefore $|\overrightarrow a.\overrightarrow b|=\mid|\overrightarrow a||\overrightarrow b|\cos\theta\mid$
$|\overrightarrow a\times\overrightarrow b|=\mid|\overrightarrow a||\overrightarrow b|\sin\theta\mid$
But it is given $|\overrightarrow a.\overrightarrow b|=|\overrightarrow a\times\overrightarrow b|$
Step 2:
Therefore $\mid|\overrightarrow a||\overrightarrow b|\cos\theta\mid=\mid|\overrightarrow a||\overrightarrow b|\sin\theta\mid$
$\Rightarrow \cos\theta=\sin\theta$
This is possible only if $\theta=\large\frac{\pi}{4}$
Therefore $\theta=\large\frac{\pi}{4}$
Hence B is the correct answer.
answered May 24, 2013