2.2 Vectors in three dimensions  (Page 3/14)

Before moving on to the next section, let’s get a feel for how ${ℝ}^3$ differs from ${ℝ}^2.$ For example, in ${ℝ}^2,$ lines that are not parallel must always intersect. This is not the case in ${ℝ}^3.$ For example, consider the line shown in [link] . These two lines are not parallel, nor do they intersect.

You can also have circles that are interconnected but have no points in common, as in [link] .

We have a lot more flexibility working in three dimensions than we do if we stuck with only two dimensions.

Writing equations in ℝ ³

Now that we can represent points in space and find the distance between them, we can learn how to write equations of geometric objects such as lines, planes, and curved surfaces in ${ℝ}^3.$ First, we start with a simple equation. Compare the graphs of the equation $x=0$ in $ℝ,{ℝ}^2,\text{and}{ℝ}^3$ ( [link] ). From these graphs, we can see the same equation can describe a point, a line, or a plane.

In space, the equation $x=0$ describes all points $\left(0,y,z\right).$ This equation defines the yz -plane. Similarly, the xy -plane contains all points of the form $\left(x,y,0\right).$ The equation $z=0$ defines the xy -plane and the equation $y=0$ describes the xz -plane ( [link] ).

Understanding the equations of the coordinate planes allows us to write an equation for any plane that is parallel to one of the coordinate planes. When a plane is parallel to the xy -plane, for example, the z -coordinate of each point in the plane has the same constant value. Only the x - and y -coordinates of points in that plane vary from point to point.

As we have seen, in ${ℝ}^2$ the equation $x=5$ describes the vertical line passing through point $\left(5,0\right).$ This line is parallel to the y -axis. In a natural extension, the equation $x=5$ in ${ℝ}^3$ describes the plane passing through point $\left(5,0,0\right),$ which is parallel to the yz -plane. Another natural extension of a familiar equation is found in the equation of a sphere.