Equations of Lines — Question 9

Question 9

Show that $$L_1:\vec r=\left\langle 2,-1,3\right\rangle+t\left\langle 4,2,-2\right\rangle\quad\text{and}\quad L_2:\vec r=\left\langle 8,2,0\right\rangle+s\left\langle-2,-1,1\right\rangle$$ describe the same geometric line. Find the parameter conversion between $t$ and $s$.

Question 9 – Solution

Strategy Check parallel directions and show that one base point lies on the other line.

Same point set The second direction is $-\tfrac 12\left\langle 4,2,-2\right\rangle$. Also $\left\langle 8,2,0\right\rangle=\left\langle 2,-1,3\right\rangle+\tfrac 32\left\langle 4,2,-2\right\rangle$, so the second base point lies on $L_1$. Thus the lines coincide.

Parameter conversion Equating the representations gives $$\vec r=\vec r_0+t\vec d=\vec r_0+\frac 32\vec d-\frac{s}{2}\vec d,$$ so $\boxed{t=\frac 32-\frac{s}{2}}$, equivalently $s=3-2t$.

Verification The conversion makes all three coordinates identical.