Equations of Lines — Question 1

Question 1 – Solution

Strategy Subtract the given points to obtain a direction, reduce it, and set one coordinate at a time equal to zero to find the coordinate-plane traces.

Line equations Since $B-A=\left\langle 6,4,-8\right\rangle=2\left\langle 3,2,-4\right\rangle$, $$\boxed{\vec r=\left\langle -2,1,3\right\rangle+t\left\langle 3,2,-4\right\rangle},$$ $$x=-2+3t,\qquad y=1+2t,\qquad z=3-4t,$$ $$\boxed{\frac{x+2}{3}=\frac{y-1}{2}=\frac{z-3}{-4}}.$$

Coordinate-plane traces Setting $z=0$, $x=0$, and $y=0$ gives, respectively, $$\boxed{L\cap xy=\left(\tfrac 14,\tfrac 52,0\right)},\quad \boxed{L\cap yz=\left(0,\tfrac 73,\tfrac 13\right)},\quad \boxed{L\cap xz=\left(-\tfrac 72,0,5\right)}.$$

Verification These correspond to $t=3/4$, $2/3$, and $-1/2$ in the same parametric equations, so all three points lie on the line.