Precalculus  Formula Sheet – Vectors  Chapter 11
                 Magnitude of a vector v = ai + bj                                                               = 
                 Vector v with initial point P  = (x ,y ) and terminal point P  = (x , y ) is equal to point is equal to the 
                                                                               1           1     1                                                   2            2      2
                 position vector v = (x -x )i + (y -y )j = ai + bj
                                                                  2     1              2     1
                 Unit vector:                              
                 Writing a vector in terms of magnitude and direction                                                                               v =                                                             or <|v|cos,|v|
                 sin>
                 Definition of Dot Product                                                                a a  + b b
                                                                                                               1 2            1 2
                 Angle Between Two Vectors       cosθ= ∣u⋅v∣
                                                                                                                  ∥u∥⋅∥v∥
                 Vectors in Space
                 Distance Formula
                                                                                                          2                       2                       2
                                                                              d=√(x −x ) +(y −y ) +(z −z )
                                                                                             2        1              2        1              2        1
                 Midpoint Formula
                                                                                                                     x +x             y +y             z +z
                                                                                                          M=( 1                 2 ,      1       2 ,     1        2)
                                                                                                                          2                2                2
                 Equation of a Sphere
                                                                                                                      2                   2                   2        2
                                                                                                       (x−h) +(y−k) +(z− j) =r
                 Cross Product of two vectors u and v:
                                   u x v = (u v  - u v )i - (u v  - u v )j + (u v  - u v )k
                                                                 2 3           3 2                1 3          3 1                 1 2           2 1
                 Where
                                                  i      j      k
                                   u x v =   u                          u         u
                                                                  1        2         3
                                                 v                       v         v
                                                                   1       2       
                                                                                    sinθ= ∥u×v∥
                 Angle of a vector  in space                                                    ∥u∥⋅∥v∥