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378                                                                                Movie Scripts


                            In fact this is a system of linear equations whose solutions form a plane with
                            normal vector (1, 2, 5). As an augmented matrix the system is simply

                                                             1 2 5  3 .

                            This is actually RREF! So we can let x be our pivot variable and y, z be
                            represented by free parameters λ 1 and λ 2 :
                                                          x = λ 1 ,  y = λ 2 .

                            Thus we write the solution as
                                                       x   = −2λ 1   −5λ 2  +3
                                                       y   =   λ 1
                                                       z   =          λ 2
                            or in vector notation
                                                                          
                                                   x      3         −2        −5
                                                  y   =   0   + λ 1   1   + λ 2   0   .
                                                   z      0          0          1
                            This describes a plane parametric equation. Planes are ‘‘two-dimensional’’
                            because they are described by two free variables. Here’s a picture of the
                            resulting plane:





























                            The Story of Your Life

                            This video talks about the weird notion of a ‘‘length-squared’’ for a vector
                                                 2    2   2
                            v = (x, t) given by ||v|| = x − t used in Einstein’s theory of relativity. The

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