Multivariable calculus and vector analysis

A line or a plane or a point?

Example 1: $y=3x-2$

Is the graph of $y=3x-2$ the graph of a line or a plane or a point? You might recognize this as a familiar form of the equation of a line with slope 3 and $y$-intercept $-2$, as in the following graph.

However, if we think $y=3x-2$ is a line, we have implicitly assumed that the equation $y=3x-2$ is an equation in two dimensions, i.e., in the two variables $x$ and $y$. It could actually be an equation in three dimensions, say with the variables $x$, $y$, and $z$. Although the $z$ doesn’t appear in the equation, we could think of the equation as having a $z$ with a coefficient of zero. We could write the equation as $y=3x+0z-2$. Here’s what plotting the equation $y=3x-2$ in a three-dimensional coordinate system looks like.

It basically looks like the previous graph. But it is really the equation of a plane. To see that, take your mouse, click on the above graph, and drag the mouse around while holding the mouse button down. This will rotate the axes, and you can see that the graph is actually a plane. (Press the Home key to return to the original view.)

We conclude that we don’t know what the graph of $y=3x-2$ is unless we specify the number of dimensions. If we want to specify a line, one way is to use a parametrization of a line. The parametrization of a line has the advantage that it gives you a line no matter how many dimensions you are working in.

Example 2: $x=3$

Is the graph of $x=3$ the graph of a line or a plane or a point? Just like above, the answer will depend on the dimensions we are working in. In one dimension, $x=3$ is just a point, as in a point on the number line.

In two dimensions, the graph is a line, as you can see by rotating the below graph. (Drag your mouse on the picture as before. We really want to just rotate the graph upward, but it can get confusing because you can rotate it in any direction. )

And, in three dimensions, the graph of $x=3$ is a plane. (Rotate the following graph to see.)

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

1b. Lines and planes

• Parametrization of a line
• Forming planes
• A line or a plane or a point?
• Parametrization of a plane
• Distance from point to plane
• Parametrization of a line examples
• Forming plane examples
• Plane parametrization example
• Distance from point to plane example
• Intersecting planes example

1c. Matrices

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material