Let's check.

Project Rho givesXg = -(0.0550*X) -(0.8732*Y) - (0.4839*Z)

Yg = (0.4940*X) - (0.4449*Y) + (0.7470*Z)

Zg = -(0.8677*X) - (0.1979*Y) + (0.4560*Z)

And in matrix form that's

**Code:**

⎡X[sub]g[/sub]⎤ ⎡ -0.0550 , -0.8732 , -0.4839 ⎤⎡X⎤

⎢Y[sub]g[/sub]⎥ = ⎢ 0.4940 , -0.4449 , 0.7470 ⎥⎢Y⎥

⎣Z[sub]g[/sub]⎦ ⎣ -0.8677 , -0.1979 , 0.4560 ⎦⎣Z⎦

*

(We curse Nyrath under our breath for using capital X, Y, and Z instead of lowercase, because ordinates are conventionallly lowercase and a position vector is conventionally

**X**. We decide to use

**A** for the position vector to avoid confusion.)

If

**Ag** =

**R** **A**Then we can premultiply both sides by

**R**-1**R**-1 **Ag** =

**R**-1**R** **A** **R**-1**R** is the appropriate-dimensioned identity matrix

**I3** by definition, so

**R**-1 **Ag** =

**A**i.e.

**A** =

**R-1** **Ag**Right?

So we plug

**Code:**

⎡ -0.0550 , -0.8732 , -0.4839 ⎤

⎢ 0.4940 , -0.4449 , 0.7470 ⎥

⎣ -0.8677 , -0.1979 , 0.4560 ⎦

into the matrix inverter at

http://matrix.reshish.com/inverse.php, check the input (especially signs), press "calculate" and get

**Code:**

-0.05505281429661579 0.4940301836677352 -0.8677206228901982

-0.8735900485746186 -0.4450385588930613 -0.19799653730732772

-0.48388771398704145 0.7469229376393793 0.4559107473356467

And then we get very suspicious because that looks a lot more like

**R**T than we expected. Is the inverse of a rotation matrix its transpose? Could be. Try another matrix inverter. Here's one:

http://ncalculators.com/matrix/inverse-matrix.htmType the original rotation matrix R into that and press "calculate", get:

**Code:**

-0.0548 , 0.4939 , -0.8683

-0.8742 , -0.4449 , -0.1979

-0.4838 , 0.7468 , 0.4558

There is an explanation that this gadget is using an analytical solution for 3×3 matrices in which it calculates a matrix called the "adjoint" and then divides by the determinant of the original matrix as explained

in Wikipedia. (I'm not familiar with these analytical special cases: I was taught matrix algebra by statisticians.) Now, the determinant for a rotation matrix ought to be 1 or -1 exactly, and it gets something damned close. The thing it calls the Adjoint (which I haven't heard of) looks very close to the transpose, but perhaps that's correct for a rotation matrix.

Okay, so we do a web search on "rotation matrix tranpose inverse" and discover that yes indeed, the inverse of any rotation matrix is simply its own transpose. If we had known that at the beginning we could have saved a lot of trouble. We have learned something, but we feel rather stupid. In our own defence, we are an economist, and economists are calculus guys, not algebraic geometry guys.

<sigh>

Okay, so our original transformation was

**Code:**

⎡X[sub]g[/sub]⎤ ⎡ -0.0550 , -0.8732 , -0.4839 ⎤⎡X⎤

⎢Y[sub]g[/sub]⎥ = ⎢ 0.4940 , -0.4449 , 0.7470 ⎥⎢Y⎥

⎣Z[sub]g[/sub]⎦ ⎣ -0.8677 , -0.1979 , 0.4560 ⎦⎣Z⎦

and because it is a rotation its inverse is

**Code:**

⎡X⎤ ⎡-0.0550 , 0.4940 , -0.8677 ⎤⎡X[sub]g[/sub]⎤

⎢Y⎥ = ⎢-0.8732 , -0.4449 , -0.1979 ⎥⎢Y[sub]g[/sub]⎥

⎣Z⎦ ⎣-0.4839 , 0.7470 , 0.4560 ⎦⎣Z[sub]g[/sub]⎦

Spreading that out into a trio of separate equations for the co-ordinates individually

X = -0.0550 X

g + 0.4940 Y

g + -0.8677 Z

gY = -0.8732 X

g + -0.4449 Y

g + -0.1979 Z

gZ = -0.4839 X

g + 0.7470 Y

g + 0.4560 Z

gIs that what I had before? No, it isn't. The most likely explanation is that I made a typing error entering the co-efficients of the matrix into the on-line matrix inversion tool.

__________________________________________

* Given what "code" tags do to formatting, it would be nice to have a monospace font to do this sort of thing with.