# Intersection of Two Lines Using Vectors

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The intersection of two lines, each given by a pair of points (i.e. the respective equations are not used), is obtained by elementary vector considerations.

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Contributed by: Tomas Garza (January 2012)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The method relies on *Mathematica*'s capabilities to handle vectors and the angles between them. If is the angle between the two lines, and is the angle between the red segment and the line* * (see step 2 in the figure), then it can readily be seen that the position vector of the point of intersection is

,

(, implying that the two lines are not parallel), where and are the position vectors of the points and ,* * is the length of the red segment, and is a unit vector in the direction of the line , from to .

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