Question

Is vector product associative?

Answer 1

The vector product of two vectors is the product of magnitude of vectors, the sine of angle between them and unit vector perpendicular to the plane of vectors. The associative property says that while applying a mathematical operation, the sequence of numbers does not matter.

Complete step-by-step solution:
Vector product or cross product of two vectors is the product of the magnitude of the two vectors, the sine of angle between them and the unit vector perpendicular to the plane of the vectors. Therefore, let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors then the vector product between them is given by-
$\overrightarrow{a}\times \overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\sin \theta \widehat{n}$
Here, $\left| \overrightarrow{a} \right|$ is the magnitude of $\overrightarrow{a}$
$\left| \overrightarrow{b} \right|$ is the magnitude of $\overrightarrow{b}$
$\theta$ is the angle between the vectors
$\widehat{n}$ is the unit vector perpendicular to the plane of the vectors
The direction of the vector product is determined using the right hand thumb rule which states that if the fingers are rotated from the first vector to the second then the thumb represents the direction of the perpendicular. By this rule, if we apply associative property and flip the vector product as $\overrightarrow{b}\times \overrightarrow{a}$
Then the direction of the perpendicular will be opposite to the original vector product due to which the value of $\widehat{n}$ changes and hence the value of the vector product also changes. Hence, the vector product is not associative.
Therefore, the vector product is not associative as the direction of perpendicular changes by right hand thumb rule.

Note: The cross product of two vectors can also be calculated using the determinant formula. The scalar product of two vectors is the product of magnitude of vectors and the cosine of angle between them. The vector product of two parallel vectors is zero while for two perpendicular products, it is maximum.