# 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…