Two lines in three-dimensional space are skew if they are neither parallel nor intersecting.

For example, consider:

Their direction vectors ⎝⎛​123​⎠⎞​ and ⎝⎛​210​⎠⎞​ are not scalar multiples of each other, so the lines are not parallel. If you try to set r1​=r2​ to find an intersection, you get a system of equations that yields no consistent solution:

Setting r1​=r2​ yields the system:

Equating components:

From 3+3λ=0, we get λ=−1. Substituting into 1+λ=2μ gives 1−1=0=2μ⟹μ=0.

Then the second equation becomes 2+2(−1)=1+0⟹0=1, a direct contradiction.

Since no (λ,μ) pair satisfies all equations, the lines do not intersect, confirming they are skew.