How do I find the unit tangential vector and the unit normal vector?

Question

I'm given that a particle travels counter-clockwise on a circle centered at the origin with radius 2. It lies in the $xy$-plane and I am also given that $r(-4) = \langle\sqrt{2},\sqrt{2},0\rangle$.

I know that the unit normal vector and unit tangential vectors are perpendicular the entire time because it is a circle and has a constant speed.

I also know that the $z(t)$ component for the two vectors will be $0$. I just don't know how to find the $x(t)$ and $y(t)$ components

I need to figure out what $T(-4)$ and $N(-4)$ are (unit tangential vector and unit normal vector).

Answer 1

For a circle, the normal vector is in the direction along the radius. Depending on your convention, you can have normal vector pointing "in" or "out": $$N(-4)=\pm\left(\frac{\sqrt 2}2,\frac{\sqrt 2}2,0\right)$$ The tangential vector, as you mentioned, is in the $xy$ plane, and it's perpendicular to $N$. You can either use the $N\times\hat z$ cross product, or just rotate $N$ by $90^\circ$. It is important to note that $T$ is not only perpendicular to $N$, but also it is along the direction of motion. So $$N(4)=\left(-\frac{\sqrt 2}2,\frac{\sqrt 2}2,0\right)$$