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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?
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In the $yz$-plane, $y+z=10$ is the equation of a line. In $xyz$-space, it's the equation of a plane. In a $4$-dimensional space, it's the equation of a $3$-dimensional affine subspace. And so on..... – Michael Hardy Apr 26 '12 at 12:33
Yes, it is the equation for a plane because it is the same as $0x+1y+1z=10$. – Henning Makholm Apr 26 '12 at 12:40
A plane that intersects the $x,y,z$-axes at $(a,0,0),(0,b,0),(0,0,c)$ is given by the equation $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$. Since the equation $y+z=10$ is equivalent to $0x+\frac{y}{10}+\frac{z}{10}=1$, we can say that the point $(a,0,0)$ lies at infinity, i.e. the plane is parallel to the $x$-axis. – Américo Tavares Apr 26 '12 at 14:39

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.

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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 $ax+by=c$.

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