Dot Product of Column Matrices

The dot product is also defined for the column matrices that represent two vectors. For two-dimensional column matrices the dot product is defined as:

Let a = ( a₁, a₂ )^T
Let b = ( b₁, b₂ )^T

Then the dot product is:

a · b = a₁b₁ + a₂b₂

Multiply corresponding elements of each column matrix, then add up the products. The result is a scalar value.

Sometimes the dot product of column matrices is written like this: a^T b (but it is defined the same way). The reason for this second, odd notation will be apparent in a later chapter when matrix multiplication is discussed.

Here is an example:

a   =   ( 1, 2 )^T

b   =   ( 3, 4 )^T

a · b  =  1*3 + 2*4  =  3 + 8  =  11