NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ and $|\vec{a} \times \vec{b}| = |\vec{a} \cdot \vec{b}|$ , then $\theta$ is equal to:

Choose the correct answer:

Explanation

1. Recall the definitions:

The magnitude of the cross product is: $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta$
The absolute value of the dot product is: $|\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| \cos \theta$

2. Set up the equation based on the given condition:

|\vec{a} \times \vec{b}| = |\vec{a} \cdot \vec{b}|

|\vec{a}| |\vec{b}| \sin \theta = |\vec{a}| |\vec{b}| \cos \theta

3. Simplify the equation:

Assuming $|\vec{a}| \neq 0$ and $|\vec{b}| \neq 0$ :

\sin \theta = \cos \theta

4. Solve for $\theta$ :

Divide both sides by $\cos \theta$ :

\frac{\sin \theta}{\cos \theta} = 1

\tan \theta = 1

Since $\theta$ is the angle between two vectors, it typically lies in the range $[0, \pi]$ . The value for which $\tan \theta = 1$ in this range is:

\theta = \frac{\pi}{4}

Conclusion:

The angle between the vectors is $\pi/4$ .

The correct option is (d).