3.15 Relationships between linesIB Maths: Analysis and Approaches HL: Revision notes

• Coincident: the same line (parallel directions and a common point).
• Parallel: directions are scalar multiples, but no common point.
• Intersecting: not parallel, and they meet at exactly one point.
• Skew: not parallel and they never meet. This can only happen in three dimensions.

In two dimensions, two lines that are not parallel always intersect.

1. Compare the direction vectors. If one is a multiple of the other, the lines are parallel: test whether a point of one line lies on the other. Yes means coincident, no means parallel and distinct.
2. If not parallel, equate the position vectors using different parameters ($\lambda$ and $\mu$) and solve two of the component equations.
3. Substitute into the third equation. If it holds, the lines intersect; if not, they are skew.

Once $\lambda$ (or $\mu$) is found and checked in all three equations, substitute it back into its line to get the coordinates. Example: $L_1:\ \mathbf{r}=(1,0,2)+\lambda(1,2,-1)$ and $L_3:\ \mathbf{r}=(2,5,0)+\nu(1,p,3)$ meet only if the $y$ equation holds too. The $x$ and $z$ equations give $\nu=\frac14$, $\lambda=\frac54$, and the $y$ equation then forces $p=-10$. The point is $\left(\frac94,\frac52,\frac34\right)$.

When $\mathbf{r}=\mathbf{a}+t\mathbf{v}$ describes motion, two paths may cross while the objects are never at the crossing point together. To test for a collision, use the same $t$ for both and solve $\mathbf{r}_A(t)=\mathbf{r}_B(t)$.

For $\mathbf{r}_A=(1,2,4)+t(2,1,0)$ and $\mathbf{r}_B=(-1,15,0)+t(3,-2,1)$, the paths cross at $(11,7,4)$, but $A$ is there at $t=5$ and $B$ at $t=4$, so there is no collision.