Lines and Planes, Part 1

Lines and Planes in Three Dimensions

Part 1: Lines in Three Dimensional Space

Lines in the plane are familiar to most people with a high school geometry background. Lines in space take these same objects and allow them to be placed in any location in three dimensional space. Just as with lines in the plane, two points determine a line in space. We'll start with this idea and end up with the symmetric equation of a line.

Vector Equations

Two points in space, P₀ = (x₀, y₀, z₀) and P = (x, y, z), form a line segment P₀P that is parallel to some vector v. We can then write P₀P = tv for some real number t. This vector v points in the direction of the line we want and it can also be expressed as the difference of the two vectors r₀ and r determined by the points P₀ and P, respectively.

This gives us the equation

r₀ - r = tv

Alternatively, we can write this as r = r₀ + tv which expresses the points on our line in terms of the given point r₀ and a given vector v. This is the vector equation of a line L parallel to the vector v and passing through the point r₀.

Write down the coordinates of such a point r using the given point r₀ = (x₀, y₀, z₀) and a given vector v = (a, b, c). The resulting coordinates are the parametric equations of the line.
Write the vector equation and the parametric equations of the line passing through the points (-1,2,1) and (5,1,0).

Symmetric Equations

Another useful interpretation of the parametric equations results from recalling that if each coordinate of the vector v is non-zero, then we can solve for t in each of the parametric equations. From this we get a set of equalities

which defines a line in space.

Write down the symmetric equations for the line you found in step 2.

Parallel, Intersecting and Skew Lines

We've already seen how vectors in space can be parallel. Lines L₁ and L₂ in space will also be parallel precisely when the vectors v₁ and v₂ that define the lines are parallel.

Similarly, two lines will intersect if they are not parallel and we can solve the equations gotten by setting each coordinate equal for a value of t that makes both lines pass through the same point.

Lines that are neither parallel nor intersecting in space are called skew. Skew lines are identified by eliminating the first two possibilities.

Determine whether the lines L₁, parallel to v₁ = (0,1,1) and passing through (1,2,3), and L₂, parallel to v₂ = (1,1,0) and passing through (1,-1,1), are parallel, intersecting or skew.

As we will see in the next section, vectors can also be used to describe planes in space.