# 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 and .

R=UW

1

R=VZ

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