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
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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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