The position vectors and direction vectors for two lines are given:
Line
L1:r=a+tb, where
a=−+3 and
b=2−+λ.
Line
L2:r=c+sd, where
c=− and
d=+2−.
To determine whether the lines are coplanar or skew, we set up equations based on their parametric forms:
L1=()+t()L2=()+s()By equating the components of the parametric equations, we get:
1+2t=s−1−t=2s3+λt=−1−sSolving the first two equations simultaneously gives:
2+4t=2s−1−t=2s=From these equations, solve for
t and
s :
⇒t=−⇒s=− after substituting
t.
For the third equation:
3−=−1+From this, solve for
λ :
21−=⇒λ=Thus, the lines are skew when
λ≠. If
λ=, the lines would be coplanar.