Dot product

The dot product of two vectors a = [a1, a2, ... , an] and b = [b1, b2, ... , bn] is defined as:

$\mathbf {a} \cdot \mathbf {b} =\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$

where Σ denotes summation notation and n is the dimension of the vector space.

In dimension 2, the dot product of vectors [a,b] and [c,d] is ac + bd. Similarly, in a dimension 3, the dot product of vectors [a,b,c] and [d,e,f] is ad + be + cf. For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is

$[1,3,-5]\cdot [4,-2,-1]=(1\times 4)+(3\times -2)+(-5\times -1)=4-6+5=3$.

The dot product can also be obtained via transposition and matrix multiplication as follows:

$\mathbf {b} ^{\mathrm {T} }\mathbf {a} =\mathbf {a} ^{\mathrm {T} }\mathbf {b}$,

where both vectors are interpreted as column vectors, and aT denotes the transpose of a, in other words the corresponding row vector.