How do you find a vector parametric equation r(t) for the line through points P = ( − 3 , − 1 , 1 ) and Q = ( − 8 , − 4 , 5 )&lt; If r ( 6 ) = P and r ( 10 ) = Q ?

Question

How do you find a vector parametric equation r(t) for the line through points   P  =  (  −  3  ,     −  1  ,     1  ) and   Q  =  (  −  8  ,     −  4  ,     5  )&amp;lt; If   r  (  6  )  =  P and   r  (  10  )  =  Q ?

Scarlet Reid · Accepted Answer

Step 1The general form of the 3D vector equation of a line is a point plus a vector multiplied by a scalar, t:      r          (      t      )        =          (              x        p            ,              y        p            ,              z        p            )        +    t          (              x        v            ,              y        v            ,              z        v            )      The parametric equations are:      x    =    t          x      v        +          x      p            y    =    t          y      v        +          y      p            z    =    t          z      v        +          z      p      Equations (1) and (2) are the x parametric equation evaluated at 6 and 10 respectively:      -    3    =    6          x      v        +          x      p                     [1]            -    8    =    10          x      v        +          x      p                     [2]      Eliminate             x      p       by subtracting equation (1) from equation (2)      -    5    =    4          x      v      Solve for             x      v       :            x      v        =    -          5      4      Substitute       -          5      4       for             x      v       in the general equation:      r          (      t      )        =          (              x        p            ,              y        p            ,              z        p            )        +    t          (      -              5        4            ,              y        v            ,              z        v            )      Substitute for             x      v       in equation (1):      6          (      -              5        4            )        +          x      p        =    -    3  Solve for             x      p       :            x      p        =          9      2      Substitute             9      2       for             x      p       in the general equation:      r          (      t      )        =          (              9        2            ,              y        p            ,              z        p            )        +    t          (      -              5        4            ,              y        v            ,              z        v            )      Equations (3) and (4) are the y parametric equation evaluated at 6 and 10 respectively:      -    1    =    6          y      v        +          y      p                     [3]            -    4    =    10          y      v        +          y      p                     [4]      Subtract (3) from (4):      -    3    =    4          y      v      Solve for             y      v                  y      v        =    -          3      4      Substitute into the general equation:      r          (      t      )        =          (              9        2            ,              y        p            ,              z        p            )        +    t          (      -              5        4            ,      -              3        4            ,              z        v            )      Substitute       -          3      4       for             y      v       in equation (3):      -    1    =    6          (      -              3        4            )        +          y      p      Solve for             y      p       :            y      p        =          7      2      Substitute into the general equation:      r          (      t      )        =          (              9        2            ,              7        2            ,              z        p            )        +    t          (      -              5        4            ,      -              3        4            ,              z        v            )      Equations (5) and (6) are the z parametric equation evaluated at 6 and 10 respectively:      1    =    6          z      v        +          z      p                     [5]            5    =    10          z      v        +          z      p                     [6]      Subtract equation (5) from equation (6):      4    =    4          z      v      Solve for             z      v                  z      v        =    1  Substitute into the general equation:      r          (      t      )        =          (              9        2            ,              7        2            ,              z        p            )        +    t          (      -              5        4            ,      -              3        4            ,      1      )      Substitute 1 for             z      v       in equation (5):      1    =    6    +          z      p                  z      p        =    -    5  Substitute into the general equation:      r          (      t      )        =          (              9        2            ,              7        2            ,      -      5      )        +    t          (      -              5        4            ,      -              3        4            ,      1      )      The above is the vector equation.The parametric equations are:      x    =    -          5      4        t    +          9      2            y    =    -          3      4        t    +          7      2            z    =    t    -    5  The symmetric equations are:                    x        -                  9          2                            -                  5          4                      =                  y        -                  7          2                            -                  3          4                      =                  z        -        5            1

How do you find a vector parametric equation r(t) for the line through points P = ( &#x

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