Breaking down the equation of a plane

Question

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.

Answer 1

One way to think about this is to realize that a plane is made up of infinitely many parallel lines side by side. It's easy to think about lines in 2D space, and 3D space is just infinitely many 2D-planes laid vertically, infinitesimally next to each other.

So when $z=0$, we would have the line $3x+y=7$, which is just a line with y-intercept at $(0,7)$ and x-intercept at $(\frac{7}{3}, 0)$, etc.

When $z=1$, we would have a line that is shifted towards the origin: $3x+y=6$, but parallel to the original $3x+y=7$ line. We would get the same thing for $z=2,3$, or in general any real number $z=c$ there would be an infinite number of parallel lines getting shifting towards the origin as we go up the $z$-axis. Hence, we get a tilted plane composed of infinitely many $3x+y=c$ lines.

Here is what the graph actually looks like.

Answer 2

Consider the collection of vectors $\{\vec x:\vec x\perp (3\vec i+\vec j+\vec k)\}$. The endpoints of these form a plane through the origin. If you shift this plane upwards $7$ units, you get the plane in question.

Answer 3

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.