Question 206413
<pre><font size = 4 color = "indigo">

I did this problem for you recently, Nick.  Here it is again.
Is there something about it that you didn't understand about
lines in space?  Would you understand it better if I used
the <b><i>i</b></i>,<b><i>j</b></i>,<b><i>k</b></i> notation instead of the ‹a,b,c› notation? 

If you need further help, you may email me at AnlytcPhil@aol.com

A line parallel to the vector <b>v</b> = ‹a,b,c› and passing through the
point P({{{x[1]}}},{{{y[1]}}},{{{z[1]}}}) is represented by the
parametric equations

{{{x=x[1]+at}}}, {{{y=y[1]+bt}}}, {{{z=z[1]+ct}}}

or as the symmetric equations:

{{{(x-x[1])/a=(y-y[1])/b=(z-z[1])/c}}}

if none of a,b, or c are 0.

Begin by using the points P(2,1,-3) and Q(-1,4,1) 
to find a direction vector for the line passing through 
P and Q, given by
    __
<i><b>v</b></i> = PQ = ‹-1-(2),4-1,1-(-3)› = ‹-3,3,4›

So we substitute in 

{{{x=x[1]+at}}}, {{{y=y[1]+bt}}}, {{{z=z[1]+ct}}}

with ‹a,b,c› = ‹-3,3,4›  and the point P({{{x[1]}}},{{{y[1]}}},{{{z[1]}}}) = P(2,1,-3)

{{{x=2-3t}}}, {{{y=1+3t}}}, {{{z=-3+4t}}}

That's the parametric equations for the line.

If you want the symmetric equation of the line, we
substitute in

{{{(x-x[1])/a=(y-y[1])/b=(z-z[1])/c}}}

{{{(x-2)/(-3)=(y-1)/3=(z-(-3))/4}}}

{{{(x-2)/(-3)=(y-1)/3=(z+3)/4}}}

Edwin</pre>