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Dot Product Basics

The main thing to understand is that the Dot Product is a process that takes 2 vectors of equal number of elements, and returns a single scalar number representing the correlation between the 2 input vectors.

The dot product comes up in many areas of math, such as within linear algebra, and it comes out here. Here we will go over the algebraic interpretation which is easier to literally understand, and the geometric interpretation, which grants more insight

Algebra Interpretation

Given 2 vectors with the same number of elements, multiplying each likewise element of each vector, then summing the results of those into a single number results in the dot product.

or in other words, given vectors <1,2> and <3,4>, the dot product between them would be:

(1*3) + (2*4) = 11 

Geometry Interpretation

States that the Dot Product is the product of vector magnitudes scaled by the angle between them.

Important things to note about this:

  • The dot product is 0 when the angle between vectors is 90 degrees, regardless of vector length
  • The dot product sign is dependent on the angle between the vectors. 
    • an acute angle (less than 90 degrees) results in a positive result
    • an obtuse angle (above 90 degrees) results in a negative result

In essence a Dot product becomes more positive the more the vectors point in the same direction, will be 0 if if they are perpendicular, and will be negative if they are pointing away

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