Finding the Unit Normal Vector at t=0: A Headache-Inducing Problem

[seeingstars63]

The position vector is r(t)= \sqrt{2}t\bar{i}+e^{t}\bar{j}+e^{-t}\bar{k} and it is asking for the unit normal vector at the parameter which is t=0. I have tried this problem many times and I guess it is all of the simplifying that is driving me bonkers!



2. I know that to find the tangent vector you would use T(t)=\frac{r'(t)}{magnitude of r'(t)}. From there, it gets crazy because you have to find the derivative and the magnitude of the tangent vector and pop it into the formula to find the Normal vector which is N(t)=\frac{T'(t)}{magnitude of T'(t)}. Then you just substitute 0 for all t's in the equation and get the unit normal vector.



3. I did not get very far in this problem and looking back at my work makes me get a head ache. If anyone can offer any assistance with this problem, that would be great. Thanks:]

 

[Mamooie312]

Normal vector (dot) Tangent vector = 0 usually if you don't care about the binormal ones.

 

[seeingstars63]

Oh, I wasn't familiar with that formula, but for this one, I have to find the derivative of the position vector and put it in the tangent vector formula, and then from there, I have to put the derivative of the tangent vector into the normal vector formula. I'm hoping that someone can help me with this problem.