# Coplanarity of Two Lines

#### notes

Coplanarity of Two Lines be
vec r = vec a_1 + lambda vec b _1               ...(1)
and vec r = vec a _2 + mu vec b _2              ...(2)
The line (1) passes through the point, say A, with position vector vec a_1 and is parallel to vec b _1. The line (2) passes through the point , say B with position vector vec a_2 and is parallel to vec b_2.
Thus , vec (AB)= vec a _2 - vec a_1
The given lines are coplanar if and only if vec (AB) is perpendicular to vec b_1 xx vec b_2.
i.e. vec (AB) . (vec b _1 xx vec b_2) = 0  or
(vec a_2 - vec a_1) . (vec b_1 xx vec b_2) = 0

Cartesian form
Let (x_1, y_1, z_1) and (x_2, y_2, z_2) be the coordinates of the points A and B respectively.
Let a_1, b_1, c_1 and a_2, b_2, c_2 be the direction ratios  of vec b _1 and vec b _2 ,  respectively.  Then
vec (AB) = (x_2 - x_1) hat i + (y_2 - y_1) hat j + (z_2 - z_1) hat k
vec b_1 = a_1 hat i + b_1 hat j + c_1 hat k   and
vec b_2 = a_2 hat i + b_2 hat j + c_2 hat k
The given lines are coplanar if and only if  vec (AB) . (vec b_1 xx vec b_2) = 0 . In the cartesian form , it can be expressed as
|(x_2 - x_1 , y_2 - y_1 , z_2 - z_1), (a_1 , a_1 , a_1), (a_2 , b_2 , c_2)| = 0   ...(4)

If you would like to contribute notes or other learning material, please submit them using the button below.

#### Video Tutorials

We have provided more than 1 series of video tutorials for some topics to help you get a better understanding of the topic.

Series 1

Series 2

### Shaalaa.com

Coplanarity of Two Lines [00:03:02]
S
0%