It is defined as:

Work is the dot product of force and displacement.

- the dot product is commutative, i.e.
**a**·**b**=**b**·**a**. - two non-zero vectors
**a**and**b**are perpendicular if and only if**a**·**b**= 0 - the dot product is bilinear, i.e.
**a**·(*r***b**+**c**) =*r*(**a**·**b**) + (**a**·**c**)

From these it follows directly that the dot product of two vectors **a** = [*a*_{1} *a*_{2} *a*_{3}] and **b** = [*b*_{1} *b*_{2} *b*_{3}] given in coordinates can be computed particularly easily:

**a**·**b**= a_{1}b_{1}+ a_{2}b_{2}+ a_{3}b_{3}

**a**·**b**=**a****b**^{T}

The dot product satisfies all the axioms of an inner product. In an abstract vector space, the notion of angle between the elements of the space can be *defined* in terms of the inner product.

;Consider a vector :**v** =v_{1}**i** + v_{2}**j** + v_{3}**k**.
;Repeated application of the Pythagorean theorem determines that :v^{2} =(v_{1}^{2} + v_{2}^{2} + v_{3}^{2}).
;which is the same formula for the dot product of the vector **v** with itself, thus :**v**·**v** = v^{2}.

;Now consider two vectors **a** and **b** from the origin and separated by an angle θ. A third vector **c** may be defined as :**c** = **a**-**b**.
;Using the law of cosines, we have :c^{2} = a^{2} + b^{2} - 2.a.b.cos(θ).
;And substituting the dot product for the squared lengths, we get :**c**·**c** = **a**·**a** + **b**·**b** - 2.a.b.cos(θ).
;But as **c** = **a** - **b**, we also have : **c**·**c** = (**a** - **b**)·(**a** - **b**).
;which expands to :**c**·**c** = **a**·**a** + **b**·**b** - 2.**a**·**b**.
;Then merging the two **c**·**c** equations we obtain :**a**·**a** + **b**·**b** - 2.**a**·**b** = **a**·**a** + **b**·**b** - 2.a.b.cos(θ).
;Subtracting **a**·**a** + **b**·**b** from both sides leaves :- 2.**a**·**b** = - 2.a.b.cos(θ).
;And dividing by -2 derives the final :**a**·**b** = a.b.cos(θ).

*See also:* Cross product