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