Finding the x and y components of a vector that is projected tangent to a circle

Question

Here is a diagram I made for the problem.

There is a very similar question on this website, but there wasn't a clear solution for that one when it came to the components of the vector. They seemed to just find the magnitude of that vector as seen here: Finding the component of a vector tangent to a circle

The problem is how do I find the actual x and y components of this tangent vector because all of the solutions I've seen don't find signed components which I need for the simulation I am programming. I took vector calculus and linear algebra about 3 years ago, so I just can't seem to figure out this problem.

Answer 1

It looks like you’re looking for the projection of the green vector onto the tangent to the large circle. This is the same as finding the orthogonal rejection from the normal to the circle at that point, which is the difference between the vector and its projection onto the normal.

Let $\vec n=\langle x_g-x_b,y_g-y_b\rangle$ and $\vec v$ be the velocity vector. $\vec n$ is parallel to the radius of the large circle at the green circle’s location, so is clearly normal to it there. The tangential velocity is then $$\vec v_{\text{tan}}=\vec v-{\vec v\cdot\vec n\over\vec n\cdot\vec n}\vec n.$$ Taking the values in the illustration, this gives $$\vec n=\langle128,196\rangle-\langle64,64\rangle=\langle64,132\rangle \\ \vec v=\langle10,3\rangle \\ \vec v_{\text{tan}} = \langle10,3\rangle-{\langle10,3\rangle\cdot\langle64,132\rangle\over\langle64,132\rangle\cdot\langle64,132\rangle}\langle64,132\rangle=\langle10,3\rangle-{1036\over21520}\langle64,132\rangle\approx\langle6.92,-3.35\rangle.$$

Answer 2

Follow these steps:

1) Construct a unit vector $\vec n$ perpendicular to $BG$ ($B$ and $G$ are the centers of blue and green circles):

$$ \vec n={(-y_G+y_B,x_G-x_B)\over\sqrt{(x_G-x_B)^2+(y_B-y_G)^2}}. $$

2) If $\vec v$ is the velocity vector (green) then the vector you want (orange) is:

$$ (\vec n\cdot\vec v)\vec v. $$