Find the orgin of a coordinate system relative to another?

I have 2 coordinate systems, Q and U

I have a point, p, that I have the coordinates for in each system ie I have (Qpx, Qpy) and (Upx, Upy).

I also have the angle between Qx-axis and Ux-axis, theta.

Do I have enough information to calculate the Q-coordinates of the origin of U?

If so how do I go about this? If not what else do I need?

I should add that I'm interested in following cases if they are different:

1) the length of the unit basis vectors e_{Q} and e_{U }are equal

2) the length of the unit basis vectors e_{Q} and e_{U }are not equal

Apologies if posted the wrong sub forum.

Re: Find the orgin of a coordinate system relative to another?

I think you could request that this thread be moved to University Math Help Forum > Advanced Algebra, which covers linear algebra, and the tool I would use in order to attempt to solve this problem.

Here, I don't claim to write a solution, only what I've considered in an attempt to help solve the problem. I haven’t taken a linear algebra course in a very long time.

Supposing coordinate system Q and U are in R^2

"I also have the angle between Qx-axis and Ux-axis, theta." At which point does the Q x-axis and U x-axis intersect? (The case for "which points" would be when the angle theta is 0)

I think a diagram would also be useful because I also don't know what you mean by "x-axis" in this context. In R^2 there is only one x-axis. If it's between a specific pair of basis vectors, one for Q and one for U, specify them

You also have coordinates for point p in Q and U.

In the usual Cartesian coordinate plane, coordinates represent the linear combination of the orthonormal unit bases (1,0) and (0,1).

Without definition for what coordinates are for Q and U, I can only go so far as to say that these represent linear combinations of basis vectors for Q and U (not necessarily orthonormal unit bases)

If you represent p as a vector, then you can create a system of equations, where a's and b's are scalars and q's and u's are basis vectors used to generate coordinates in Q and U systems respectively.

p = a_1 * q_1 + a_2 * q_2

p = b_1 * u_1 + b_2 * u_2

I believe the real crux to the solution is to use projection (with given angle theta at a point) and re-write each basis vector for coordinate system U as a linear combination of basis vectors for coordinate system Q.

As for your cases, I don't think the length of basis vectors matter too much. This would only change the scalars, a's and b's, when representing a vector such as p in the span of basis vectors of either U or Q.

What would make a vector a unit vectors is if the norm of the vector is equal to 1.