Normal Direction of a Plane

In a d-dimensional space,

Example: The following equation defines a plane in 3-D space spanned by $x$, $y$ and $z$:

  $\displaystyle f(x,y,z)=ax+by+cz+d=x+2y+3z+4=0
\nonumber
$  
where $a=1$, $b=2$, $c=3$ and $d=4$. Solving for $z$ we get
  $\displaystyle z=g(x,y)=-\frac{1}{c} ( ax+by+d )
=-\frac{1}{3}(x+2y+4)
\nonumber
$  

When $x=y=0$, we get $z=-c/d=-4/3$, the intercept of the plane on z-axis. The normal direction of the plane is composed of the coefficients of the three variable $x$, $y$ and $z$ in $f(x,y,z)$:

  $\displaystyle {\bf n}=[a,\,b,\,c]^T=[1,\;2,\;3]^T
$ (425)
or the coefficients of the two variables $x$ and $y$ of $g(x,y)$, with the last component $-1$:
  $\displaystyle {\bf n}=\left[-\frac{a}{c},\,-\frac{b}{c},\,-1\right]^T
=-\frac{1}{c}[a,\,b,\,c]^T=-\frac{1}{3}[1,\;2,\;3]
$ (426)