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WOLFRAM|DEMONSTRATIONS PROJECT

Intersection of Two Lines Using Vectors

step
1
2
3
4
original setting
adjust the position
of point U
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.
step 1
Objective: to find the intersection of the two lines
R
1
=UW
and
R
2
=VZ
.
step 2
The projection (blue line
VY
) of the red segment
UV
through
V
perpendicular to
R
1
onto
R
1
has magnitude
r=|UV|sin(α)
, where
α
is the angle between the red segment and
R
1
.
step 3
In vector terms, the tip of the blue vector
VY
is at a distance
d=|UV|cos(α)
from
U
, so that its position vector
Y
is
U±dH
, where
H
is a unit vector (in purple). Use the positive sign whenever
α<π/2
; otherwise use the negative sign.
step 4
Let
β
be the angle betwen the lines
R
1
and
R
2
. The point of intersection of the two lines (in orange) lies at the distance
e=r/sin(β)
from
V
(with
sin(β)0
), so that its position vector is
V±eK
, where
K
is a unit vector (in black). Choose the sign to give the position closest to
U
.
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