Are these planes?

Question

Given the equation: y + z = 10

• Can it be considered a plane? Why (not)?
• How do you correctly express planes which are normal onto one axis, for example a plane that lies completely vertical in space or a plane that lies completely horizontal in space?

Answer 1

You know how to express a line that is horizontal or vertical in the plane? Horizontal lines are given by $y=c$, vertical by $x=c$, right? Similar thing for planes in 3-dimensional space. Horizontal planes are $z=c$. Vertical planes are $ax+by=c$.

Answer 2

In general a plane in $\mathbb{R}$^3 is given by an equation of the form $$ ax + by + cz = d.$$

If for example $a= 0$ you see that the equation is "independent" of $x$. So that means that given a point in the plane, you can vary $x$ as you like and still move in the plane. Or said differently, if you travel parallel to the $yz$-plane, you stay in the plane. So the plane is normal (perpendicular) to the $x$-axis.

Likewise if $b = 0$ the plane is normal to the $y$-axis.

And if $c = 0$ the plane is normal to the $z$-axis.