Could I get some help on this vector value function?

[dmalwcc89]

Homework Statement

A fighter plane, which can only shoot bullets straight ahead, travels along the path r(t) = <5 - t, 21 - t^2, 3 - (t^3/27)>. Show that there is precisely one time t at which the pilot can hit a target located at the origin.

Homework Equations

I think I am supposed to use a form of the equation for vector parametrization for a tangent line at r(t) = L(t) = r(t) + t[r'(t)].

The Attempt at a Solution

I computed the derivative of r(t): r'(t) = <-1, -2t, -(1/9)t^2>. My book shows examples, but they all have a t value to compute the equation at. I don't quite understand how to find a line that goes through the origin that is tangent to where the fighter is straight ahead, since I don't know that spot either. I graphed it on a 3D parameter graph so I could visualize what is going on, but how do I find these settings?

 

[Dick]

If the pilot can hit the origin the vector r(t) and the vector r'(t) must be parallel and pointed in opposite directions. So r(t)=k*r'(t). Write out that vector equation, giving you three equations in k and t, pick two of them and solve for t.

 

[dmalwcc89]

If the pilot can hit the origin the vector r(t) and the vector r'(t) must be parallel and pointed in opposite directions. So r(t)=k*r'(t). Write out that vector equation, giving you three equations in k and t, pick two of them and solve for t.

Thank you very much. I was able to solve the problem, but for study purposes, what is the significance of setting a k in there?

 

[Dick]

Thank you very much. I was able to solve the problem, but for study purposes, what is the significance of setting a k in there?

Just expressing that one vector is a constant multiple of the other. That's what it means to be parallel.

 

[dmalwcc89]

Just expressing that one vector is a constant multiple of the other. That's what it means to be parallel.

Perfect. Thank you very much for your help.