5. Vectors

d. Dot Product

1. Algebraic Definition

If \(\vec u=\left\langle u_1,u_2\right\rangle\) and \(\vec v=\left\langle v_1,v_2\right\rangle\) are vectors, then their dot product or scalar product or inner product is the scalar \[ \vec u\cdot\vec v=u_1v_1+u_2v_2 \] Thus, the dot product is the sum of the pairwise products of the corresponding components of the two vectors. Memorize this!

The name dot product is used to emphasize that it is denoted by putting a dot between the two vectors.

The name scalar product is used to emphasize that the result is a scalar rather than a vector, unlike vector addition, which takes two vectors and gives back a third vector.

The name inner product is used to indicate that it is a special case of a more general product that you will learn about it in a course on linear algebra.

If \(\vec u=\left\langle 1,2\right\rangle\) and \(\vec v=\left\langle -2,5\right\rangle\), compute \(\vec u\cdot\vec v\).

\(\vec u\cdot\vec v=(1)(-2)+(2)(5)=-2+10=8\)

The meaning of the dot product will become clear when we get to its geometrical interpretation and its applications. However, the derivation of the geometrical interpretation requires several properties of the dot product which we first discuss on the next page.