How do you convert ( y = 2 + x ) to parametric equation?

Question

How do you convert             (      y      =      2      +      x      )       to parametric equation?

lywiau63 · Accepted Answer

Step 1The general line equation in             ℝ      2       is      l    →    a    x    +    b    y    +    c    =    0  this equation can be arranged as      l    →          〈              n        →            ,      p      -              p        0            〉        =    0   where            n      →        =          {      a      ,      b      }       a line generic point, and             p      0        =          {              x        0            ,              y        0            }       a line given point.The determination of             p      0       is straightforward, comparing      a    x    +    b    y    +    c    =    0   with      a          (      x      -              x        0            )        +    b          (      y      -              y        0            )        =    0   concluding      -    a          x      0        -    b          y      0        =    c  then supposing that       b    ≠    0   and fixing             x      0        =    0   we get             y      0        =    -          c      b       wo            p      0        =          {      0      ,      -              c        b            }      Now, the parametric representation.We have a line point which is             p      0       and an orthogonal vector             n      →       to the line direction. We need a vector             v      →       in the line direction, so choosing             v      →       such that             ∥                        v          →                    ∥        &amp;gt;    0   and             〈              n        →            ,              v        →            〉        =    0  . Choosing             v      →        =          {      b      ,      -      a      }       then             〈              n        →            ,              v        →            〉        =    b    a    -    a    b    =    0  .Finally, the parametric line equation is      l    →    p    =          p      0        +    λ          v      →      Step 2In the present case      l    →    -    x    +    y    -    2    =    0              p      0        =          {      0      ,      -              c        b            }        =          {      0      ,      -              (                              -            2                    1                )            }                  n      →        =          {      -      1      ,      1      }        →          v      →        =          {      1      ,      1      }      then       l    →          {                                    x            =            0            +            λ                                                y            =            2            +            λ

How do you convert <mstyle displaystyle="true"> ( y = 2

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