Notes taken from the Linear Algebra YouTube series by 3Blue1Brown.
Vectors
“The introduction of numbers as coordinates is an act of violence.” -Hermann Weyl
There are two main views by that of a physics student and one of a CS student:
- Physics student: A vector is defined by length and direction; an arrow pointing in space.
- CS student: A vector is a structured list of numbers of which its length determines its dimension.
Key Takeaway: It would be foolish to only look at vectors in the classical coordinate sense, as that would obscure countless domains of problems meant to be solved. Rather, they express a sort of relationship.
Dot Product
A common operation for which the intuition is difficult to understand. In essence, the dot product is a scalar value that represents “how much are these two vectors facing the same way.” The dot product of two vectors facing the same direction will be positive. If facing opposite directions, they will be negative. Lastly, if they are exactly perpendicular the dot product will be 0.
Calculating is simple, multiply each component of the vectors and add the products together. In the case of a 2D coordinate system, this would look like x1 * x2 + y1 * y2.
Common Transformations
Perpendicular/Reverse Vector
Easiest way to imagine this besides memorizing the rules is to consider a vector of [1, 0]. If we wanted to turn 90° CCW, we would get [0, 1]. For 90° CW we would get [0, -1].