Question

What if the cross product is $\overrightarrow{0}$?

Answer 1

Cross Product:

Cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both vectors.

Formula: The vector product or cross product of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\overrightarrow{\mathbf{A}}\mathbf{\times }\overrightarrow{\mathbf{B}}\mathbf{=}\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\hat{\mathbf{n}}$

If the cross product of two vectors is $\overrightarrow{0}$, then

• Both vectors are parallel to each other.
• The angle between the vectors is $\overrightarrow{0}$.
• One of them or both vectors are zero vectors.

Example: The cross product of two vectors $\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{B}=2\hat{i}+2\hat{j}+2\hat{k}$ is,

$\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 2& 2& 2\end{array}\right|$

$\Rightarrow$ $\overrightarrow{A}\times \overrightarrow{B}=\left(1·2-2·1\right)\hat{i}-\left(1·2-2·1\right)\hat{j}+\left(1·2-2·1\right)\hat{k}$

$\Rightarrow$$\overrightarrow{A}\times \overrightarrow{B}=\left(2-2\right)\hat{i}-\left(2-2\right)\hat{j}+\left(2-2\right)\hat{k}$

$\Rightarrow$$\overrightarrow{A}\times \overrightarrow{B}=0\hat{i}-0\hat{j}+0\hat{k}$

$\Rightarrow$$\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{0}$

Hence, if cross product of any two vectors is zero then vectors are parallel, one vector or both vectors are zero.