Dot Product

The dot product (or inner product) of two vectors $\left(\vphantom{ a_{1},a_{2},...,a_{n}}\right.$a1, a2,..., an$\left.\vphantom{ a_{1},a_{2},...,a_{n}}\right)$ and $\left(\vphantom{
b_{1},b_{2},...,b_{n}}\right.$b1, b2,..., bn$\left.\vphantom{
b_{1},b_{2},...,b_{n}}\right)$ is defined by

$\displaystyle \left(\vphantom{ a_{1},a_{2},...,a_{n}}\right.$a1, a2,..., an$\displaystyle \left.\vphantom{ a_{1},a_{2},...,a_{n}}\right)$$\displaystyle \left(\vphantom{
b_{1},b_{2},...,b_{n}}\right.$b1, b2,..., bn$\displaystyle \left.\vphantom{
b_{1},b_{2},...,b_{n}}\right)$ = a1b1 + a2b2 + ... + anbn

The dot is available on the Common Symbols toolbar and in the Binary Operations drop-down panel below itbpF0.3009in0.3009in0.0701inbinop.wmf.

For the following examples of dot products with n = 3, define

a = $\displaystyle \left[\vphantom{
\begin{array}{ccc}
1 & 2 & 3
\end{array}
}\right.$$\displaystyle \begin{array}{ccc}
1 & 2 & 3
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{ccc}
1 & 2 & 3
\end{array}
}\right]$, b = $\displaystyle \left(\vphantom{
\begin{array}{r}
1 \\
0 \\
-1
\end{array}
}\right.$$\displaystyle \begin{array}{r}
1 \\
0 \\
-1
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{r}
1 \\
0 \\
-1
\end{array}
}\right)$, c = [3, 2, 1], and d = (2, - 1, 0)

with New Definition from the Define submenu.

$\blacktriangleright$ Evaluate

$\left(\vphantom{ u,v,w}\right.$u, v, w$\left.\vphantom{ u,v,w}\right)$$\left(\vphantom{ x,y,z}\right.$x, y, z$\left.\vphantom{ x,y,z}\right)$ = ux + vy + wz

(1, 2, 3)⋅(3, 2, 1) = 10

ac = 10

ab = - 2

cd =  4