Geometry

Shape formulas

Figure             Perimeter     Area            Volume   

Triangle           l1 + l2 + l3   (1/2)bh
Rectangle          2l + 2w       lw
Cirlce             πd            πr2
Parallelogram                    bh                   
Trapezoid                        (1/2)h(l1 + l2)           

Sphere                           4πr2            (4/3)πr3
Cone                             πrs(side)       (1/3)πr2h
Cylinder                         2πrh(side)      πr2h
Pyramid                          2sb(sides)      (1/3)b2h
Rectangular Solid                                lwh

Where:
 ln is the length of a side
 b is the length of the base
 h is the perpendicular height
 l is the length
 w is the width
 d is the diameter
 r is the radius
 s is the ?

Vectors

Definition

Examples

• velocity
• force
• displacement

Properties

initial point  - tail of arrow (start)
terminal point - tip of arrow (end)
components     - the coordinates of the vector written as v = (v1, v2, ..., vn-1)
               - terminal point when initial point is at zero
equal          - when length and direction are the same, position
                 doesn't matter.
norm           - length or distance between the initial and terminal points
               - norm of vector v is written as ||v||
unit vector    - when norm is 1
normal vector  - a vector perpendicular to a plane; therefore it
                 is restricted to 3-space

Vector arithmetic

u + v = (u1 + v1, u2 + v2, ..., un-1 + vn-1)
u - v = (u1 - v1, u2 - v2, ..., un-1 - vn-1)
kv = (kv1, kv2)

Where:
 u and v are vectors
 k is a scalar

Theorems

u + v = v + u
(u + v) + w = u + (v + w)
u + 0 = 0 + u = u
u + (-u) = 0
k(lu) = (kl)u
k(u + v) = ku + kv
(k + l)u = ku + lu
1u = u

Where:
 u and v are vectors
 k and l are scalars

Norm of a vector (length)

Definition

||u|| = √(u12 + u22)

||u|| = √(u12 + u22 + u32)

Dot product (multiplying vectors)

Definition

If u and v are vectors in 2-space or 3-space and θ is the angle between u and v,
then the dot product or Euclidean inner product u·v is defined by:

      ì ||u|| ||v|| cos θ     if u ≠ 0 and v ≠ 0
u·v = í
      î 0                     if u = 0 and v = 0

Derived definition

u·v = u1v1 + u2v2 + ... + un-1vn-1

Finding the angle between vectors

If u and v are nonzero vectors, then:

           u·v
cos θ = ———————————
        ||u|| ||v||

Theorems

v·v = ||v||2
If the vectors u and v are nonzero and θ is the angle between them, then:
        θ is acute      if and only if     u·v > 0
        θ is obtuse     if and only if     u·v < 0
        θ = π/2         if and only if     u·v = 0
u·v = v·u
u·(v + w) = u·v + u·w
k(u·v) = (ku)·v = u·(kv)
v·v > 0 if v ≠ 0, and v·v = 0 if v = 0

Cross product (normal vector)

Definition

If u = (u1, u2, u3) and v = (v1, v2, v3) are vectors in 3-space,
then the cross product u × v is the vector defined by:

u × v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)

Theorems

u·(u × v) = 0                         (u × v is orthogonal to u)
v·(u × v) = 0                         (u × v is orthogonal to v)
||u × v||2 = ||u||2||v||2 - (u·v)2     (Lagrange's identity)
u × (v × w) = (u·w)v - (u·v)w         (relationship between cross and dot products)
(u × v) × w = (u·w)v - (v·w)u         (relationship between cross and dot products)
u × v = -(v × u)
u × (v + w) = (u × v) + (u × w)
(u + v) × w = (u × w) + (v × w)
k(u × v) = (ku) × v = u × (kv)
u × 0 = 0 × u = 0
u × u = 0

Where:
 u, v and w are vectors in 3-space
 k is a scalar

Misc

• The plural for vertex is vertices.
  
• Normal means perpendicular or V · U = 0.
  
• Ortho means perpendicular or V · U = 0.
  
• Orthogonal means perpendicular or V · U = 0.
  
• "Ortho" in orthonormal means that (1,0) . (0,1) = 0, and
  "normal" means that (1,0) . (1,0) = 1 and (0,1) . (0,1) = 1.
  In other words, the dot product of any two basis vectors is zero if
  you've chosen two different basis vectors, and 1 if you're taking the
  dot product of any basis vector with itself.