Question

Why is Cross Product not Commutative?

Answer 1

Cross Product

1. The cross-product of two vectors is an operation of mathematics that is performed on two vectors.
2. The outcome of the cross product is a vector that is perpendicular to both vectors and has a magnitude that is equal to the magnitude of the first vector times the magnitude of the second vector times $\sin \left(\theta \right)$ where $\theta$ is the angle between the two vectors.

The resultant of the cross product gives two can give two directions that are perpendicular to the original vectors and are in the exact opposite directions. To find which of these two directions the cross product uses, we will use the right-hand rule.

Right-Hand Rule

1. According to the right-hand rule, in your right hand, point your index finger in the direction of the first vector and your middle finger in the direction of the second vector, and hold your thumb up.
2. Now, your thumb is indicating the direction of the cross-product vector.

Explanation

1. Now if we switch the order of the input vectors then the resultant vector will point exactly in the opposite direction.
2. Cross product is not commutative because the order of vectors matters in the cross product.

Hence the cross product is not commutative.