Useful Linear Algebra

Dot Product

• Find the interior angle between two vectors

$$A\cdot B=|A||B|cos\theta$$

$$\theta =\arccos ({\displaystyle \frac{A\cdot B}{|A||B|}})$$

$$whereA\cdot B=A_x\times A_y+B_x\times B_y$$

Cross Product

• Get perpendicular vector from two vectors
• Find area created by two vectors (for 2D, parallelogram area, for 3D, parallelepiped area)
  • This is the magnitude of perpendicular vector

Matrix Multiplication

There is a formula for calculating the resultant matrix from a matrix multiplication.

$$\left[\begin{array}{cc}{X}_{22}{Y}_{11}+X_{12}{Y}_{21}& X_{22}{Y}_{12}+X_{12}{Y}_{22}\\ X_{11}{Y}_{21}+X_{21}{Y}_{11}& X_{11}{Y}_{22}+X_{21}{Y}_{12}\end{array}\right]$$

But it’s much more meaningful to think of a matrix multiplication of multipling matrix A by each column of matrix B.

Similar how you would transform a vector by multiplying it by the matrix, you’re transforming each component of the matrix which describes the coordinate space.