Coplanarity of Two Lines


 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)

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