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

In summary, to find the unit normal vector at t=0 for the position vector r(t)= \sqrt{2}t\bar{i}+e^{t}\bar{j}+e^{-t}\bar{k}, you would first use the formula T(t)=\frac{r'(t)}{magnitude of r'(t)} to find the tangent vector. Then, using the formula N(t)=\frac{T'(t)}{magnitude of T'(t)}, you would find the normal vector. Finally, substituting t=0 into the equation would give you the unit normal vector. However, this process can be complicated and may require assistance.
  • #1
seeingstars63
8
0
The position vector is r(t)= [tex]\sqrt{2}[/tex]t[tex]\bar{i}[/tex]+e[tex]^{t}[/tex][tex]\bar{j}[/tex]+e[tex]^{-t}[/tex][tex]\bar{k}[/tex] 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)=[tex]\frac{r'(t)}{magnitude of r'(t)}[/tex]. 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)=[tex]\frac{T'(t)}{magnitude of T'(t)}[/tex]. 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:]
 
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  • #2


Normal vector (dot) Tangent vector = 0 usually if you don't care about the binormal ones.
 
  • #3


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.
 

What is a unit normal vector?

A unit normal vector is a vector that is perpendicular to a given curve or surface at a specific point. It has a magnitude of 1 and is used in vector calculus to calculate curvature, normal force, and other physical quantities.

Why is finding the unit normal vector at t=0 a headache-inducing problem?

Finding the unit normal vector at t=0 can be a headache-inducing problem because it involves solving complex mathematical equations and requires a deep understanding of vector calculus. It also requires a lot of time and patience to arrive at the correct solution.

How is the unit normal vector calculated at t=0?

The unit normal vector is calculated at t=0 by first finding the derivative of the given curve or surface at t=0. Then, this derivative is divided by its magnitude to obtain a unit vector. This unit vector is the unit normal vector at t=0.

What is the significance of finding the unit normal vector at t=0?

Calculating the unit normal vector at t=0 is important in many applications, such as in physics, engineering, and computer graphics. It helps determine the direction and strength of forces acting on an object at a specific point, and is used in calculating motion, acceleration, and other physical quantities.

What are some tips for solving the headache-inducing problem of finding the unit normal vector at t=0?

Some tips for solving this problem include breaking it down into smaller steps, using geometric intuition, and practicing with simpler examples. It is also helpful to have a strong understanding of vector calculus and to double-check all calculations to ensure accuracy.

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