NIMCET 2012 — Mathematics PYQ

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Question

NIMCET 2012 — Mathematics PYQ

NIMCET | Mathematics | 2012

If $\theta (0 \le \theta \le \pi)$ is the angle between the vectors $\vec{a}$ and $\vec{b}$ , then $\frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}}$ is equal to:

Choose the correct answer:

Explanation

1. Recall the Formulas:

Magnitude of Cross Product: The magnitude of the cross product $\vec{a} \times \vec{b}$ is given by:

$|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta$
Dot Product: The dot product $\vec{a} \cdot \vec{b}$ is given by:

$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$

2. Substitute into the Given Expression:

We need to find the value of $\frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}}$ :

\frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}} = \frac{|\vec{a}| |\vec{b}| \sin \theta}{|\vec{a}| |\vec{b}| \cos \theta}

3. Simplify the Expression:

The magnitudes $|\vec{a}|$ and $|\vec{b}|$ cancel out from the numerator and the denominator:

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

Conclusion:

The expression $\frac{|\vec{a} \times \vec{b}|}{\vec{a} \cdot \vec{b}}$ simplifies to $\tan \theta$ .

Correct Option:

(b) $\tan \theta$

Choose the correct answer:

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