# How to determine if two 3D vectors intersect?

## Question:

How to determine if two 3D vectors intersect?

## Intersections:

To intersect two curves, we equalize the variables of each of the curves: if there are no parameter values that solve all the equalities, we will say that the curves do not intersect.

Two intersects two curves given in the 3D vectors equation, we must equalize the variable of the curves.

For example, given the curves: {eq}{r_1}\left( t \right) = \left\langle {t,t + 1,t} \right\rangle ,\quad {r_2}\left( s \right) = \left\langle {s + 1,s + 2,1} \right\rangle {/eq}, equalizing the variables, we have:

{eq}{r_1}\left( t \right) = \left\langle {t,t + 1,t} \right\rangle ,\quad {r_2}\left( s \right) = \left\langle {s + 1,s + 2,1} \right\rangle \\ \\ \left\{ \begin{array}{l} t = s + 1\\ t + 1 = s + 2\\ t = 1 \end{array} \right. \to t = 1,s = 0 {/eq}

So, in this case, the curves intersect at the point {eq}\left( {1,2,1} \right). {/eq}