NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

Let $\vec{a}$ and $\vec{b}$ be two vectors. Which of the following vectors are not perpendicular to each other?

Choose the correct answer:

Explanation

1. Key Concept: The Cross Product

The cross product $\vec{a} \times \vec{b}$ results in a vector that is perpendicular to the plane containing both $\vec{a}$ and $\vec{b}$ . This means:

$(\vec{a} \times \vec{b}) \cdot \vec{a} = 0$
$(\vec{a} \times \vec{b}) \cdot \vec{b} = 0$

2. Evaluating the Options

Option (a): As stated above, $\vec{a} \times \vec{b}$ is always perpendicular to $\vec{a}$ .
Option (b): Since both $\vec{a}$ and $\vec{b}$ lie in the same plane, their sum $(\vec{a} + \vec{b})$ also lies in that plane. Therefore, $(\vec{a} + \vec{b})$ is perpendicular to the cross product $\vec{a} \times \vec{b}$ .
Option (d): Similarly, the difference $(\vec{a} - \vec{b})$ lies in the same plane as $\vec{a}$ and $\vec{b}$ . Therefore, it is also perpendicular to $\vec{a} \times \vec{b}$ .

3. Analyzing Option (c)

Let's check the dot product of $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$ :

(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b}

Since dot product is commutative ( $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ ), the middle terms cancel:

= |\vec{a}|^2 - |\vec{b}|^2

For these vectors to be perpendicular, the result must be $0$ , which only happens if $|\vec{a}| = |\vec{b}|$ . In general, for any two arbitrary vectors, this result is not zero.

Conclusion

Vectors $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$ are not necessarily perpendicular to each other.

Correct Option: (c)