  # Dot Product (also known as Inner or Scalar Product)

The dot product is a scalar number and so it is also known as the scalar or inner product. In a real vector space, the scalar product between two vectors is computed in the following way: Besides, there is another way to define the inner product, if you know the angle between the two vectors: We can conclude that if the inner product of two vectors is zero, the vectors are orthogonal.

In Matlab, the appropriate built-in function to determine the inner product is 'dot(u,v)'.

For example, let's say that we have vectors u and v, where

u = [1  0] and v = [2  2]. We can plot them easily with the 'compass' function in Matlab, like this:

x = [1 2]
y = [0 2]
compass(x,y)

x represents the horizontal coordinates for each vector, and y represents their vertical coordinates. The instruction 'compass(x,y)' draws a graph that displays the vectors with components (x, y) as arrows going out from the origin, and in this case it produces: We can see that the angle between the two vectors is 45 degrees; then, we can calculate the scalar product in three different ways (in Matlab code):

a = u * v'
b = norm(u, 2) * norm(v, 2) * cos(pi/4)
c = dot(u, v)

Code that produces these results:
a = 2
b = 2.0000
c = 2

Note that the angle has to be expressed in radians, and that the instruction 'norm(vector, 2)' calculates the Euclidian norm of a vector (there are more types of norms for vectors, but we are not going to discuss them here).

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