We have a line in R 3 given as intersetion of two planes: { A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 How to represent it in parametric form: { x = x 0 + a t y = y 0 + b t z = z 0 + c t ?

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We have a line in         R    3   given as intersetion of two planes:      {                                        A            1                    x          +                      B            1                    y          +                      C            1                    z          +                      D            1                                              =          0                                                  A            2                    x          +                      B            2                    y          +                      C            2                    z          +                      D            2                                              =          0                        How to represent it in parametric form:      {                            x                                    =                      x            0                    +          a          t                                      y                                    =                      y            0                    +          b          t                                      z                                    =                      z            0                    +          c          t                        ?

bideanbarrenaf5 · Accepted Answer

What you look for is a point on the line       (          x      0        ,          y      0        ,          z      0        )  , and a direction vector of the line       (    a    ,    b    ,    c    )  .To find a point on the line, you can for example fix   x and find   y  ,  z (there are some case where this won&#039;t work).To find a direction vector, note that the vector       (          A      1        ,          B      1        ,          C      1        )   is orthogonal to the first plane therefore to the line. Likewise,       (          A      2        ,          B      2        ,          C      2        )   is orthogonal to the line. If you take their vector product you will get a direction vector.Another way to find a direction vector, is to find another point on the line and subtract one point from the other.

Liberty Mack · Accepted Answer

Row reduce the system of equations. Generically, you will get an equation with two variables and one with three. There is going to be a repeated variable in both, that is your parameter. Write the remaining variables in terms of this common variable and you are done.In your example:  l  :      {                            x          +          y          −          z          +          1                                    =          0                                      x          −          y          +          z          −          1                                    =          0                        Add the first equation to the last to get:  l  :      {                            x          +          y          −          z          +          1                                    =          0                                      x                                    =          0                        We can take   y or   z as the parameter. For example let&#039;s take   z. Then                    x                            =        0        +        0        z                            y                            =        −        1        +        z                            z                            =        0        +        z

We have a line in <mi mathvariant="double-struck">R 3 </msup> given as interse

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