Dot Product

  • Find the interior angle between two vectors

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

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

$$ where\ A\cdot B = A_x \times A_y + B_x \times B_y $$

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.

$$ \begin{bmatrix} X_{22} Y_{11} + X_{12} Y_{21} & X_{22} Y_{12} + X_{12} Y_{22} \\ X_{11} Y_{21} + X_{21} Y_{11} & X_{11} Y_{22} + X_{21} Y_{12} \end{bmatrix} $$

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.