Given two concurrent lines $(x,y,z)$ with a missing coordinate on the x axis , how do I find the intersection point between them?

Question

$r1 : ( x, y , z ) = (0,5,3) + t * ([a], -3, -4)$

$r2 : (x,y,z) = (0,3,−6)+ s * ([a],−1,5)$

Parametrical eq:

r1)

$ x = 0 + [a]t$

$y = 5 + (-3t)$

$z = 3 + (-4t)$

r2)

$x = 0 + [a]t$

$y = 3 - t$

$z = -6 + 5t$

I have two questions , how do I find [a] and how I get the intersection point between the lines. Thank you !

Answer 1

It seems your parametric equation r2 is not written quite right. Consider that s and t may not be the same in order to find the intersection. Setting the two y's and two z's equal gives you a linear system of equations that determines s and t (not equal). Once you have s and t, set the two x's equal to determine [a] (which is obviously 0 in this particular case.)

Once you have s, t and [a], plug them all into either of the two equations r1 or r2 to get the intersection point.