The "Dot Product" of two vectors is a scalar number that represents how much of one is in the direction of the other.

The dot product is linear in the lengths of the vectors, so for any scalar $c$ and vectors ${\bf~a}$ and ${\bf~b}$ we have $(c{\bf~a}){.}{\bf~b}=c({\bf~a}{.}{\bf~b})$

The dot product is commutative.

The geometric interpretation is that ${\bf~a}{.}{\bf~b}$ is the product of the lengths of the vectors, times the cosine of the angle between them. This implies that if the vectors are orthogonal then the dot product is zero.

If the vectors are finite dimensional then the dot product is obtained by summing the point-wise products of the coordinates.

So if ${\bf~a}=\left[\begin{matrix}a_1\\a_2\\a_3\end{matrix}\right]$ and ${\bf~b}=\left[\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right]$ then ${\bf~a}{\bf.b}=a_1b_1+a_2b_2+a_3c_3.$

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