Where Do Two Lines Intersect in 3 Dimensions?

I’m a physicist, not a mathematician. So, when we talk about straight lines in 3 dimensions I find it easier to use real examples. Don’t worry, I am going to get to those parametric and symmetric forms of 3D lines that you see in maths, but let’s start with a particle moving with a constant velocity.

Parametric Form, Symmetric Form, and Velocity Form of a 3D Line

We can define constant velocity as:

Here v is the velocity vector and r is the position vector. Suppose I know where the particle is at t = 0 seconds and I call that r0 (pretend like there is a subscript and both v and r are vectors). In that case, I can find the position (r) for any future time (t).

Note: I really don’t like setting t0 = 0, but that’s the way I can get the math-based parametric equations. There, I said it.

Since both r and v are vectors in 3D, I can write them as:

I’m using this angle-bracket notation for vectors (that’s the way all the cool physicists do it) where the three values inside the brackets are just the x, y, and z components of the vector. Yes, there are other ways to represent vectors.

Since I have a 3D vector equation, I can write that as three scalar equation — one for each component.

Boom. There are your parametric equations for a line in 3D. In this case, the parameter is t. You can find any point on the line just by changing the parameter t. Yes, you can use a negative value for t — in the real life example of velocity that would be a time before I set t = 0.

What about the symmetric form of equations? Well, all three of those equations have the same variable t. If I solve for t and then set all the t’s equal, I get:

The important thing to remember is that no matter what form you use, it’s really about a vector point (r0) and a vector direction (v).