How do you find the parametric and symmetric equation of the line passing through the point ( 2 , 3 , 4 ) and perpendicular to the plane 5 x + 6 y − 7 z = 20 ?

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How do you find the parametric and symmetric equation of the line passing through the point   (  2  ,     3  ,     4  ) and perpendicular to the plane   5  x  +  6  y  −  7  z  =  20 ?

vrtuljakwb · Accepted Answer

Step 1The vector perpendicular to the plane,       5    x    +    6    y    -    7    z    =    20        5          i      ^        +    6          j      ^        -    7          k      ^      This allows us to write the point-vector form of the line passing through the point             (      2      ,      3      ,      4      )       :            (      x      ,      y      ,      z      )        =          (      2      ,      3      ,      4      )        +    t          (      5              i        ^            +      6              j        ^            -      7              k        ^            )      From the point-vector form we can extract the 3 parametric equations by observation:      x    =    5    t    +    2        y    =    6    t    +    3        z    =    -    7    t    +    4  To find the symmetric form we solve each of the parametric equations for t and then set them equal:      t    =                  x        -        2            3            t    =                  y        -        3            6            t    =                  z        -        4                    -        7            Setting them equal gives us the symmetric form:                    x        -        2            3        =                  y        -        3            6        =                  z        -        4                    -        7

How do you find the parametric and symmetric equation of the line passing through the point (

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