Today we learn about Unit vector, vector dot product, vector cross product, triple cross product, scalar triple product.

Unit Vector. Any vector A can be represented by the magnitude of the vector |A| multiplied by its unit vector written as a_n

So A = |A| a_n.

A unit vector has the direction of the main vector is of unit magnitude. It is the ratio of vector itself by its magnitude.

Thus, the magnitude of a unit vector is one.

Scalar or Dot Product of Two Vectors:

Dot product of two vectors is a scalar quantity having the value equal to the product of the magnitudes of two vectors and the cosine of angle between them.

A and B are two vectors having an angle θ between them, the dot product between A and B is

A.B = ABcosθ

The dot product operation consists of multiplying the magnitude of one vector by the scalar obtained by projecting the second vector on to the first vector.

Laws of dot product:

The dot product operation is commutative

A.B = B.A.

Dot product also obeys distributive law,

A. (B + C) = A.B + A.C

Also A.A = A²

Vector or Cross Product of Two Vectors:.

The cross or vector product of two vectors A and B is another vector whose magnitude

is the product of magnitudes of A and B and the sine of angle theta between A and B, and whose direction is the direction of a right hand screw as it is turned from A towards B through theta.

Thus

A x B = |A||B|}sin theta

In vector product, the associative law doesn’t hold

(A x B) x C not equals to A x (B x C)

but the distributive law holds

A x (B + C) = A x B + A x C

An example of cross product is force on a current carrying conductor placed in a magnetic field,

Triple Cross Product.

A triple cross product involves three vectors and resultant is a vector.

A x (B x C) not equals to (A x B) x C

Scalar Triple Product

It involves three vectors in a dot product operation and a cross product operation that is

A.B x C = B.C x A = C.A x B

Reference: These articles are referred from my authored book “concepts of electromagnetic field theory” having ISBN 978-81-272-5245-8. Try to make the figures for products of vectors. In case of any doubt in this article or any other EMFT or physics related article, kindly post in the comment section.