Dot product

The dot product, or scalar product, is one of the two types of vector multiplication, and is widely used in many areas of mathematics and physics. Both the dot product and the cross product ( or vector product) are widely used in in the study of optics, mechanics, electromagnetism, and gravitational fields, for example.

Definition

Given two vectors, A and B, the dot product is the product of the magnitude of A, the magnitude of B and the cosine of the smaller angle between them.

A • B = |A||B|cosθ_AB

In a three dimensional cartesian coordinates, a more useful definition is

A • B = A_xB_x + A_yB_y + A_zB_z

The dot product is a scalar, not another vector, and it obeys the commutative law such that

A • B = B • A

Use in calculating Work

In mechanics, when a constant force F is applied over a straight displacement L, the work performed is FL cosθ_FL, more compactly written as the dot product below. Note in the special case the force and displacement are parallel, work = force times distance.

Work = F • L (linear motion)

Work = $\int$F • $d$L (non-linear motion)