2.10:
Scalar and Vector Triple Products
2.10:
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.
Suggested Reading
- Griffiths, D.J. (2013). Fourth Edition. Introduction to ELectrodynamics. San Francisco, CA: Pearson. pp.7.
- IDA, N. (2015). Fourth Edition. Engineering Electromagnetics. Switzerland: Springer International Publishing. pp.18.