[Coda]

The dot product of two vectors is easy. Given vectors [a_1, b_1, ...] and [a_2, b_2, ...] of the same length, the dot product is:

(a_1 * b_1) + (a_2 * b_2) + ...

Surprisingly equivalently, the dot product is ALSO the product of the vector's lengths multiplied by the cosine of the angle between them, that is, ||v_1|| * ||v_2|| * cos(θ).

Don't ask for a graphical interpretation of what this actually represents geometrically, because it doesn't have a strict meaning. And my theoretical geometry skill isn't good enough to have an intuition on when I would want to use the dot product of two vectors outside of formulas that other people have already derived.

That said, it has a lot of useful properties. For example, if the dot product of two vectors equals 0, then the vectors are perpendicular, and if two vectors are parallel, then their dot product is the product of their lengths. (The dot product of a vector with itself is therefore the square of its length, which demonstrates the Pythagorean Theorem.)