Question

State the following statement is True or False

The value of $\overrightarrow{a}\cdot (\overrightarrow{a}\times \overrightarrow{b})$ is zero.

• A. True
• B. False

Answer 1

The correct option is A True

$\overrightarrow{a}\times \overrightarrow{b}$ is defined as the cross product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$

A cross product of two vectors results in a vector perpendicular to both the vectors.

Thus, $\overrightarrow{a}\times \overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$

When taken a dot product of two vectors $\overrightarrow{p}.\overrightarrow{q}$, we get $|\overrightarrow{p}|\times |\overrightarrow{q}|\times \cos \theta$ where $\theta$ is the angle between those two vectors.

Since $\cos \left({90}^o\right)=0,\overrightarrow{a}.(\overrightarrow{a}\times \overrightarrow{b})=0$