Equations of Lines — Question 1

Question 1

Find vector, parametric, and symmetric equations of the line through $A=(-2,3,1)$ and $B=(4,-1,7)$. Determine the point on the line whose $x$-coordinate is 7.

Question 1 – Solution

Strategy Use $B-A$ as a direction vector and keep one common parameter in all coordinates.

Equations A direction is $\left\langle 6,-4,6\right\rangle=2\left\langle 3,-2,3\right\rangle$. Thus $$\vec r=\left\langle-2,3,1\right\rangle+t\left\langle 3,-2,3\right\rangle,\quad x=-2+3t,\ y=3-2t,\ z=1+3t,$$ and $\boxed{(x+2)/3=(y-3)/(-2)=(z-1)/3}$.

Requested point From $-2+3t=7$, $t=3$, giving $\boxed{(7,-3,10)}$.

Verification $t=0$ gives $A$ and $t=2$ gives $B$.