Tangent line parallel to a plane

[jakey]

Hi guys,

I'm stuck with a problem here:

Let a curve be given by the following parametric equations: x=t, y=t^2, z=t^3. At which points is the tangent line (of the curve) parallel to the plane x + 2y + z = 0?

What is the underlying principle behind this?

My thoughts:
The tangent line is parallel to the plane if their normal vectors are colinear. The normal vector of the plane is (1,2,1). Now I'm stuck here.


thanks!

 

[Some Pig]

A line is parallel to a plane if the line is perpendicular to the plane's normal.
The direction of the tangent of the curve is: (1,2t,3t^2)
So 1*1+2*2t+1*3t^2=0, 3t^2+4t+1=0, (3t+1)(t+1)=0, t=-1/3 or -1.

 

[HallsofIvy]

Hi guys,

I'm stuck with a problem here:

Let a curve be given by the following parametric equations: x=t, y=t^2, z=t^3. At which points is the tangent line (of the curve) parallel to the plane x + 2y + z = 0?

What is the underlying principle behind this?

My thoughts:
The tangent line is parallel to the plane if their normal vectors are colinear. The normal vector of the plane is (1,2,1). Now I'm stuck here.


thanks!

A line does NOT have a "normal vector" (more correctly, there are an infinite number of vectors normal to a line- a line does not have a specific "normal vector").

A line has a direction vector that must be perpendicular to the normal vector of the plane in order that the line be parallel to the plane.