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

1. ## 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?

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

1) the length of the unit basis vectors eQ and eU are equal
2) the length of the unit basis vectors eQ and eU are not equal

Apologies if posted the wrong sub forum.

2. ## 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.