NIMCET 2015 — Mathematics PYQ

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 $(\vec{a}-\vec{b})$:

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

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)