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I frequently have to solve problems that involve expressing a given plane (e.g. a horizontal plane, a vertical plane) in a Cartesian and/or vector form.

For example:

I don't know how to approach this kind of questions. Please advise on how to approach this question.

Also, given an equation, I don't always know how to give a geometric interpretation to it.

For example,

$2x-y=3$ is a plane that is parallel to the z-axis.

I don't see this and I don't know how to plot since the $z$ coordinate isn't in the equation.

How does one interpret such equations and give equations of various kinds of planes of a certain property without memorizing them?

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3 Answers

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The plane $2x - y = 3$ (or, more generally $ax + by = c$) is parallel to the $z$ axis precisely because $z$ doesn't appear in the equation. So, if some point $(x_0,y_0, z_0)$ satisfies the plane equation, then any other point of the form $(x_0,y_0, z_1)$ must also satisfy the equation, no matter what $z_1$ we choose, because we have only changed $z=z_0$ to $z=z_1$, and $z$ doesn't appear in the equation.

So, what does this mean? Well, it means that an entire infinite line of points of the form $(x_0,y_0, z_0)$ must belong to the plane. These lines are parallel to the $z$-axis, so the plane is parallel to the $z$-axis.

By this reasoning, a plane of the form $z = c$ must be parallel to both the $x$ and $y$ axes, so (if it wasn't obviously already) we see that this sort of plane must be perpenedicular to the $z$ axis.

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A horizontal plane is of the form $z=c$. The fact that $z$ doesn't appear in the equation $2x-y=3$ means that the intersection of this plane with a plane of the form $z=c$ always has the same equation (viewed as an equation in the plane $z=c$), namely $2x-y=3$. So a parametrization for such a line would be $(x=t,y=2t-3,z=c)$. In general, if you have some equation in $x$ and $y$ and want to view its solution space in $\mathbb{R}^3$, you draw what the solution space looks like in the $xy$-plane and then just extend it in the $z$ direction.

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The Cartesian coordinates $(x,y,z)$ of a point can be viewed as the components of the radius-vector $\boldsymbol{r}$ connecting the point to the origin. Take a point on the plane, $\boldsymbol{r}_0$. We look for vectors $\boldsymbol{r}-\boldsymbol{r}_0$ orthogonal to the plane's normal vector $\boldsymbol{n}$. Plane's equation is the scalar product $\boldsymbol{n}\cdot (\boldsymbol{r}-\boldsymbol{r}_0)$=0.

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