# Scalar and Vector Triple Products

JoVE Core
Physik
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JoVE Core Physik
Scalar and Vector Triple Products

### Nächstes Video2.11: Gradient and Del Operator

The scalar triple product is the dot product of a vector with the cross product of two vectors. It results in a scalar quantity.

Here, the magnitude of the cross product of two vectors gives the area of the parallelogram formed by these vectors. The resultant vector is perpendicular to the parallelogram plane.

The projection of the third vector in the direction of the resultant of the cross product gives the height of the parallelepiped.

It follows that the resultant of the scalar triple product denotes the volume of the parallelopiped formed by these vectors.

The volume remains unaltered if the vectors are rotated in cyclic order in the scalar triple product.

The Vector triple product is the cross product of a vector with the resultant of the cross product of two vectors and results in a vector quantity.

Here, if the vectors are rotated in cyclic order, the resultant vector is entirely new. So, the vector triple product is not associative.

## Scalar and Vector Triple Products

Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the resultant is a vector. These rules of the scalar or vector product between two vectors can be applied to multiple vectors to obtain meaningful combinations. The scalar triple product is the dot product of a vector with the cross product of two vectors.

The scalar triple product is the dot product of a vector with the cross product of two vectors. As before, the scalar triple product results in a scalar quantity. The scalar triple product of three vectors can be expressed as follows:

On the cyclic rotation of vectors, the result of the scalar triple product remains the same, meaning, it is associative. The scalar triple product is the projection of a vector onto the resultant of the cross product of two vectors and represents the volume defined by these three vectors.

On the other hand, the vector triple product is the cross product of a vector with the cross product of two other vectors, and it results in a vector quantity. The vector triple product of three vectors can be expressed as follows:

Here, the cyclic rotation of vectors results in a new vector. The vector triple product is not associative.