Could someone please explain how to do this step-by-step?

Let L be the line with parametric equations

x = −6−3t
y = −5−2t
z = −2−t

Find the vector equation for a line that passes through the point P=P(9, −3, −4) and intersects L at a point that is distance 4 from the point Q=Q(−6, −5, −2).

Question

Could someone please explain how to do this step-by-step?


Let L be the line with parametric equations

x = −6−3t
y = −5−2t
z = −2−t

Find the vector equation for a line that passes through the point P=P(9, −3, −4) and intersects L at a point that is distance 4 from the point Q=Q(−6, −5, −2).

e.mccormick · Answer

Not quite to this in my Lin Alg class. But I know the concept. If you are doing it through Lin Alg, this seems to discuss it pretty well: http://math.stackexchange.com/questions/348819/find-the-vector-parametric-equation-of-the-line-through-a-and-b

e.mccormick · Answer

Sorry I can't help more.  Still learning that one myself!

anonymous · Answer

I'm pretty OK with the stuff in that link, but I'm having trouble with the "point distance 4 from the point Q=Q(-6,-5,-2)" - I can't figure out how to find that point (or if that's what I'm supposed to be doing first). I assumed you would add 4 times the unit vector to Q and go from there, but that doesn't seem to be right :( Thanks for the reply!

e.mccormick · Answer

Yah, that was what stumped me right off.  Seeing it to a specific point, so it has to be a specific line, not just any particular line.

anonymous · Answer

I would figure out when the distance between $L$ and $Q$ is 4.

anonymous · Answer

$$
4^2=(L-Q) \cdot (L-Q) 
$$

anonymous · Answer

That will give you at most two $t$ values.  In this case it is supposed to only give you one though.  $L(t)$ will give you the point that is passes through.  With two points it is easy to find the vector equation.

anonymous · Answer

Awesome, thank you!