How do you write a vector equation and a parametric equation for each line: the line through A ( 1 , − 3 , 1 ) and parallel to vector u = ( 2 , − 2 , 1 ) ?

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How do you write a vector equation and a parametric equation for each line: the line through   A  (  1  ,     −  3  ,     1  ) and parallel to vector   u  =  (  2  ,     −  2  ,     1  ) ?

Elyse Huff · Accepted Answer

Step 1The general vector eqn of a line            r      →        =          a      →        +    λ          d      →                  r      →        =   the general position vector on the line            a      →        =   a known position vector on the line            d      →        =  In this case            a      →        =          (                                    1                                                -            3                                                1                              )                  d      →        =          (                                    2                                                -            2                                                1                              )      a vector eqn of the line is therefore            r      →        =          (                                    1                                                -            3                                                1                              )        +    λ          (                                    2                                                -            2                                                1                              )      For parametric eqns.            r      →        =          (                                    x                                                y                                                z                              )                  (                                    x                                                y                                                z                              )        =          (                                    1                                                -            3                                                1                              )        +    λ          (                                    2                                                -            2                                                1                              )            x    =    1    +    2    λ        y    =    -    3    -    2    λ        z    =    1    +    λ  If you want the cartesian eqn. just eliminate       λ        λ    =                  x        -        1            2            λ    =    -                  y        +        3            2            λ    =    z    -    1  cartesian eqn                    x        -        1            2        =    -                  y        +        3            2        =    z    -    1

How do you write a vector equation and a parametric equation for each line: the line through A

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