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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.

Answer and Explanation:

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}


Learn more about this topic:

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Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

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