Breaking down the equation of a plane Could someone explain the individual parts of a plane equation?
For example:
$3x + y + z = 7$
When I see this I can't imagine what it's supposed to look like.

In general a plane: $ \color{blue}{ax+by+cz=d}$ has its normal vector with direction ratios $a$, $b$ & $c$ correspoding to three orthogonal axes x, y & z respectively. A plane can also be expressed in the intercept form: $$\color{blue}{\frac{x}{\left(\frac{d}{a}\right)}+\frac{y}{\left(\frac{d}{b}\right)}+\frac{z}{\left(\frac{d}{c}\right)}=1}$$ Where, $\color{blue}{\frac{d}{a}}$, $\color{blue}{\frac{d}{b}}$ & $\color{blue}{\frac{d}{c}}$ are the intercepts cut by the plane with three orthogonal axes x, y & z respectively.    
Now, the given equation: $\color{blue}{3x+y+z=7}$ has normal vector with direction ratios $a=3$, $b=1$ & $c=1$ & cuts the intercepts $\color{green}{\frac{7}{3}}$, $\color{green}{7}$ & $\color{gree}{7}$  with three orthogonal axes x, y & z respectively.     
