Question

Show that the lines `vecr = veca + λvecb` and `vecr = vecb + μveca` are coplanar and the plane containing them is given by `vecr.(veca xx vecb) = 0`.

Answer 1

`vecr = veca + λvecb` and `vecr = vecb + μveca`

They are coplanar if shortest distance between them is zero.

S.D. = 0

S.D. = `((vec(a_2) - vec(a_1)).(vec(b_1) xx vec(b_2)))/(|vec(b_1) xx vec(b_2)|)`   ...`[vecr = vec(a_1) + λvec(b_1) and vecr = vec(a_2) + λvec(b_2)]`

S.D. = `((vecb - veca).(vecb xx veca))/(|vecb xx veca|)`

S.D. = `0/(|vecb xx veca|)`

S.D. = 0

Here, shortest distance between these lines is zero, hence the lines `vecr = veca + λvecb` and `vecr = vecb + μveca` are coplanar.