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Spec Maths Cheat Sheet - Coggle Diagram

- - - - Vector & Pythagoras
    - - Dot Product = 0 to find t
    - - Cos^-1 DP / Lengths
    - - 90 - DP of Direction and Normal
    - - Find where Point + Normal line meets plane, do 2 point from there.
        
        |ax1+by1+cz1-d|/magnitude(normal)
    - - 180 - DP of Normals
    - - Get plane from cross product between lines
  - - - R2: 0 0 0 | 0
        
        3 Coincident: Plane of Solutions
      - R2: 0 n n | n
        
        2 Coincident: Line of Solutions
        
        All 3 Meet at Line: Line of Solutions
      - R2: 0 0 0 | n
        
        2 Coincident 1 Parallel: No Solutions
    - - R2: 0 0 0 | n
        
        All 3 Parallel: No Solutions
      - R2: 0 n n | n
        
        2 Parallel 1 Intersecting: No Solutions
        
        All 3 Meet at Lines: No Solutions
    - - 3 Intersecting: Meet at a Point
  - - - Rn --> aRn
      - Rn --> Rn + aRm
      - Rn <--> Rm
    - - Reduced Row Echelon Form
        
        A 0 0, 0 B 0, 0 0 C
      - Row Echelon Form
        
        A B C, 0 D E, 0 0 F
  - - - Det(BCEF)i - Det(ACDG)j + Det(BCEF)k
  - - - [xyz] = [x1y1z1] + t[abc]
    - - x = x1+at, y=y1+bt, z=z1+ct
    - - x-x1/a = y-y1/b = z-z1/c = t
  - - - z is equal for s and t
        
        Lines intersect
      - z is not equal at s and t
        
        Lines do not intersect
      - Infinite solutions for s and t
        
        Lines are coincident
  - - - Equal
    - - Equal
    - - Equal
    - - Sum to 180
    - - Sum to 180
  - - - Overline arrow for going between points
      - Underline tilde for named vectors
- - - - Circle with origin at x,y and radius n
      - Shade in with <=. Only interior with <
    - - Perpendicular bisector of p1 and p2
    - - Ray from point z1 with angle n
  - - - nth roots of z = root(r) * cis((theta+k2pi)/n) for k = [0, ..., n-1]
        
        Sum of roots omega(0) to omega(n-1)
  - - - Positive: 2 real roots
      - Negative: No real roots
      - 0: 1 coincident root
- - - - P1 (or greater) LHS = [algebraically transform until = RHS]
        
        P1 true
        
        Assume there exists k >= 1 where Pk true i.e LHS = RHS
        
        LHS = [do the algebra again]
        
        Therefore Pk+1 true if Pk true
        
        By PMI Pn true
- - - - sqrt(x'(t)^2+y'(t)^2)
    - - Use trig on the velocity vector