Vector Geometry Review
       Vector Basics - Sections 12.1 and 12.2
     Vectors, Components and Representation
     Vectors, Scalar Multiplication
     Vectors, Addition
     Vectors, Magnitude
     Vectors, Basis
    Dot and Cross Products - Sections 12.3 and 12.4
     Vectors, The Dot Product
     Vectors, The Cross Product
      Lines and Planes - Sections 12.2 and 12.5
     Lines in 3-Space
     Planes in Space
     1 Vector Basics - Sections 12.1 and 12.2
   Vectors
        Avector is a geometric object that has magnitude (length) and direction.
        A scalar is a constant in R which has no direction, only magnitude.
        Familiar examples of vectors: force, velocity, acceleration, pressure, ŕux
        Avector can be represented geometrically by an arrow AB from A (the
                                                            ⃗    ⃗    −→
        initial point) to B (the terminal point). Notation: v = v = AB.
     Translating a vector does not                           B = (x ,y ,z )
     change it, since the magnitude and                            t  t t
                                                      ⃗             ⃗
     direction remain the same.                       v             v
     These three arrows all represent the
     same vector!
                                                            −→
                                                            AB =⟨x −x,y −y,z −z⟩
                                                                   t   i t   i t   i
       Video                                      A=(x,y,z)
                                                        i i i
   Cartesian Representation of Vectors
                             ⃗
             Draw a vector v with its initial point at the origin O.
                                    ⃗
             The components of v are the coordinates of the terminal point P.
                                       z
                                                    (2,3,1)
                                 (0,0,0)                 y
                                  x
             ⃗    −→
        Here v = OP = ⟨a,b,c⟩.
                      ⃗    −→
        In general, if v = AB where A = (x ,y ,z ) and B = (x ,y ,z ) then
                                              1  1   1              2   2  2
                              ⃗
                              v = ⟨x −x , y −y , z −z ⟩.
                                     2     1   2     1   2    1