- Points, Vectors
  - A point is a position
  - A vector is a direction and a length
    - We often use something called "normalized" vectors, which have length = 1.  This makes some of the math easier.
  - Basic arithmetic with points, vectors (P, Q, R = points, U, V, W = vectors)
    - Addition (componentwise)
      - vector + vector = vector, uses paralellogram rule
      - point + vector = point, translates the point
      - point + point = T@N#H%
        - You can't add points and have it make geometric sense
        - UNLESS! you do a weighted average, where the weights add up to 1  
        - t * point + (1 - t) * point = point
          - At t = 0, at first point
          - At t = 1, at second point
          - At t = 0.5, halfway between
          - At t = 0.25, 25% between
          - T can be negative too!
          - Called a LERP, for "linear interpolation".
    - Subtraction (componentwise)
      - vector - vector = vector, uses triangle rule
      - point - vector = point, translates the point backwards
      - point - point = vector, finds distance between points
    - Multiplication (componentwise)
      - Doesn't have any geometric meaning!
    - Multiplication (the DOT PRODUCT)
      - noted U . V = scalar (normal number)
      - Algebraic interpretation
        - U_x * V_x + U_y * V_y (in 2D)
        - U_x * V_x + U_y * V_y + U_z * V_z (in 3D)
        - Acts like you'd expect multiplication to
          - Associative     (U . V) . W = U . (V . W)
          - Commutative           U . V = V . U
          - Distributive    (U + V) . W = U. W + V . W
        - Geometric meaning
          - |U| |V| cos t
          - VERY POWERFUL
          - if t = 90deg (vectors are perpendicular), will be zero
          - if t < 90deg (vectors point in same dir), will be positive
          - if t > 90deg (vectors point in opposite dir), will be negative
          - if t = 0deg (same vectors), then V . V  = |V|^2
      - THE DOT PRODUCT IS EXTREMELY USEFUL FOR MAKING GEOMETRIC MATH SIMPLER
        - Collide Circles from HW1 -- example
          -           |(p_1 + t v_1) - (p_2 + t v_2)| = r_1 + r_2
	                |(p_1 + t v_1) - (p_2 + t v_2)|^2 = (r_1 + r_2)^2
		               |(p_1 - p_2) + t(v_1 - v_2)|^2 =

                       <expand left side>
                
	                          (p_1 - p_2).(p_1 - p_2) =
                          + 2t(p_1 - p_2).(v_1 - v_2)
                         + t^2(v_1 - v_2).(v_1 - v_2)
	
              t^2      *      (v_1 - v_2).(v_1 - v_2) = 0
	        + t        *     2(p_1 - p_2).(v_1 - v_2)
	        + (p_1 - p_2).(p_1 - p_2) - (r_1 + r_2)^2

   - Projection
     - proj_V U, gets the vector that is U's part in the direction of V
       - proj_V U = V . U
                    ----- V
                    V . V
       - Note that if V is a unit vector (of length 1), then this simplifies to (V . U) V
       - Can use this to easily calculate the distance between a point and a line

- Matrices
  - Algebraic definition
    - Use the dot product!!!
      - i'th row, j'th column is calculated with the dot product
    - Also follows MOST of the rules of multiplication
      - Associative, Distributive
      - NOT COMMUTATIVE
        - "anti-commutative", NM = trans( trans(M) trans(N) )
        
  - Geometric meaining
    - Represent linear maps, transforming space
    - You can figure out what a matrix does entirely by looking at what it does to the X and Y axes
    - Matrices to remember
      - IDENTITY MATRIX
		| 1 0 |
		| 0 1 |
	  - SCALE MATRIX
        | x 0 |
		| 0 y |
	  - ROTATION MATRIX
		| cos t	 -sin t |
		| sin t	  cos t |
	    - Note that inverse of rotation is the matrix transposed!

     - Homogenous coordinates
       - Add a "w" component, which should always be 0 or 1.  If it is not 1, divide the vector by it to make it 1.
       - Homogenous matrices to remember
         - TRANSLATION MATRIX
		   | 1 0 x |
		   | 0 1 y |
		   | 0 0 1 |
		 - PROJECTION MATRIX (useful in 3D)
		   | x 0  0  0 |
		   | 0 y  0  0 |
		   | 0 0 za zb |
		   | 0 0 -1  0 |
           - x determines X fov
           - y determines Y fov
           - za, zb together determine the near Z and far Z planes