Parametric curve, unique pt. P, tangent at P goes through other point.

[s3a]

Homework Statement

Problem:
A curve given parametrically by (x, y, z) = (2 + 3t, 2 – 2t^2, -3t – 2t^3). There is a unique point P on the curve with the property that the tangent line at P passes through the point (-10, -22, 76).

Answer:
P = (-4, -6, 22)

What are the coordinates of point P?

Homework Equations

Derivatives and vector manipulation.

The Attempt at a Solution

I read on-line that one must find some vector and that that vector is parallel to the vector of the derivatives with respect to t (so, they're scalar multiples of each other), but I don't know specifically how to start nor do I understand what is going on, so I would greatly appreciate it if someone could tell me what needs to be done to successfully complete this problem and also help me understand what is going on spatially as well.

 

[haruspex]

The point P corresponds to some value of t, t_P. You can differentiate to find dx/dt, and plugging in t = t_P gives you the tangent vector at P. From this, obtain an expression for the tangent line in terms of t_P. It remains to plug in the known point that this line passes through.