Transforming line equations into parametric/symmetric equations

Question

Yesterday I came across an exercise in my coordinate geometry book:

*Find if $a$ and $b$ coincide, are skew, parallel or intersect (if so, find the intersection point)

$$a \leftrightarrow \begin{cases} 2x + y = 4 \\ z = 3 \end{cases}$$

$$b \leftrightarrow \begin{cases} x + y = 4 \\ z = 2x \end{cases}$$ I have done many similar exercises but I have never used equations like that before. How do I transform them into parametric or symmetric equations ?

Answer 1

One fairly-automatic way to go is:

• Make one coordinate (say, $x$) equal to the parameter (say, $t$).
• Use the equations to write the other coordinates ($y$ and $z$) in terms of the parameter ($t$).

So ...

$$\begin{align} a: &\qquad x = t, \quad y = 4 - 2t, \quad z = 3\phantom{t} \quad\to\quad (x,y,z) = (0,4,3) +(1,-2,0)t \\ b: &\qquad x = t, \quad y = 4 - \phantom{2}t, \quad z = 2 t\quad\to\quad (x,y,z) = (0,4,0) + (1,-1,2)t \end{align}$$