How to quickly convert to the parametric form of equation of line?

Question

I have a straight line given by: $$ \begin{cases} x - y - 4z + 12 = 0 \\ 2x + y -2z + 3 = 0 \end{cases}$$

I'm looking to convert it to the parametric form of straight line like this:

$$\vec{r} = p_1 + t\vec{v}$$

The way I currently do it is:

1. Eliminate one variable - in this case $-3x-3y+6=0 \implies x+y=2$
2. Let one variable be $t$ - in this case $y=t$
3. Write $t$ in terms of other variables - in this case it's $t=2-x$ and $t=\frac{14-4z}{2}$
4. Write it all in one line: $\frac{x-2}{-1} = y = \frac{z-\frac{14}{4}}{- \frac{1}{2}}$
5. Write the parametric form: $\vec{r} = (2, 0, \frac{14}{4}) + t (-1, 1, -\frac{1}{2})$

First, is my method correct? Secondly, this is a rather long method and produces ugly parametric form, can I do this quicker and better?

Answer 1

You do not need step 4. you can directly settle for

$\begin{cases} x=2-t\\ y=t\\ z=\dfrac{7-t}2 \end{cases}$

And notice that you already have your parametric representation !

$M=(2,0,\frac 72)+t(-1,1,-\frac 12)$

Anyway, you can also select two values of $t$ that give "nice" points $A$ and $B$ if you are not satisfied with the fractionnal coefficients.

For instance $t=1$ gives $A=(1,1,3)$ and $t=-1$ gives $B=(3,-1,4)$

Calculate the direction vector $\vec u=\vec{AB}=(2,-2,1)$

And you get your line : $M=A+t\vec u=(1+2t,1-2t,3+t)$

Answer 2

In general form, the way you have expressed the two planes, the normal to each plane is given by the variable coefficients. If you just take the cross product of those two norms, you get the direction vector for the line. So, $$(1,-1,-4)\otimes (2,1,-2)=(6,-6,3)$$ Now you need a point that is on the line. To get that, pick one of the variables and set it to zero, then solve for the other two. That's pretty much what you did. If I let z=0, then $A=(-5,7,0)$

So the line is $$(x,y,z)=(-5,7,0)+t(6,-6,3)$$

Answer 3

The equations can be written as $$ \begin{bmatrix} 1&-1\\ 2&1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} =\begin{bmatrix} 4z-12\\ 2z-3 \end{bmatrix} $$ Invert the matrix to get $$ \begin{align} \begin{bmatrix} x\\ y \end{bmatrix} &=\frac13\begin{bmatrix} 1&1\\ -2&1 \end{bmatrix} \begin{bmatrix} 4z-12\\ 2z-3 \end{bmatrix}\\ &=\begin{bmatrix} 2z-5\\ -2z+7 \end{bmatrix} \end{align} $$ Thus, a parametric form is $$ \begin{bmatrix} x\\ y\\ z \end{bmatrix} =\begin{bmatrix} 2\\ -2\\ 1 \end{bmatrix}t +\begin{bmatrix} -5\\ 7\\ 0 \end{bmatrix} $$