Vector calculus, normals to plane curves

[miglo]

Homework Statement

show that n(t)=-g'(t)i+f'(t)j and -n(t)=g'(t)i-f'(t)j are both normal to the curve r(t)=f(t)i+g(t)j at the point (f(t),g(t)).

Homework Equations

The Attempt at a Solution

i tried finding the unit normal of r(t) in hopes that it would be exactly what n(t) and its negative are, but after going through a lot of algebra i didnt get what i was hoping would be the solution, unless i did some algebra mistake. Anyways i doubt I am approaching this problem the right way and would just like a hint at what to do.

 

[tiny-tim]

hi miglo!

hint: what is the tangent to f(t)i+g(t)j ?

 

[miglo]

f'(t)i+g'(t)j?

 

[LCKurtz]

Yes. Are the supposed normals perpendicular to the tangent?

 

[miglo]

yeah i think i just figured it out like a minute ago
i take the dot product of r'(t) with n(t) which equals 0, therefore n(t) is normal to the curve at (f(t),g(t)) and then i do the same with -n(t)