Calculating true airspeed: Horseshoe Heading Technique This question relates to calculating the speed of an aircraft relative to the air, based on GPS measurements (i.e. groundspeed measurements).  Specifically it is about David Rogers' Horseshoe Heading Technique.
The technique involves flying 3 perpendicular legs and measuring groundspeeds of $v_1, v_2, v_3$, respectively.  We are interested in calculating $v_T$, the average speed through the air (as opposed to speed over the ground), while accounting for the confounding factors of wind $v_N, v_W$ (all are scalar).
I don't understand the intuition behind the technique.  I see we can form a linear system, but I cannot see a way to solve it:
$$
\begin{pmatrix}
v_1^2 \\ v_2^2 \\ v_3^2
\end{pmatrix}
=
\begin{pmatrix}
v_T - 2v_N & v_N & v_W \\
v_T + 2v_W & v_N & v_W \\
v_T + 2v_N & v_N & v_W \\
\end{pmatrix}
\begin{pmatrix}
v_T \\ v_N \\ v_W
\end{pmatrix}
$$
So we get:
$$
\begin{pmatrix}
v_1^2 \\ v_2^2 \\ v_3^2
\end{pmatrix}
= v_T 
\begin{pmatrix}
v_T - 2v_N \\
v_T + 2v_W \\
v_T + 2v_N \\
\end{pmatrix} + v_N^2 + v_W^2 \\
$$
I'm just not sure where to go from here to actually understand these equations geometrically.
Could someone please help me understand what is going on mathematically?  While the paper appears to have a technically correct line of reasoning, there must be some way to understand this problem in a more systematic way: e.g. by recognizing ground velocity and wind velocity as a vectors and solving a system of matrix equations instead of introducing all sorts of unexplained variables (P, Q, R) and then mysteriously solving a quadratic.
 A: It is assumed that a plane flies at a constant velocity (constant engine thrust) $v_T$ with respect to the flowing air. Also the air-flow or wind, for which neither its velocity nor its direction is known, is assumed to be constant during the measurement and can be described by its a northern component $v_N$ and western component $v_W$. In order to obtain the true plane velocity also the wind velocity needs to be determined, so there are three unknown quantities $v_T,$ $v_N$, and $v_W$ and hence three equations are required. Each equation follows from a different particular flight path during which the ground speed is measured:

*

*Heading north (against northern wind and getting a drift to the east)

*Heading east (with the western wind and getting a drift to the south)

*Heading south (with the northern wind and getting a drift to the east)

These are the figures 1a,b,c in the paper. The velocities of the plane with respect to the ground are measured externally giving $v_1$,$v_2$, and $v_3$ (or equivalently the distances in each path) and expressed in term of northern and western components, giving the three equations.
Strictly speaking, any three different flight directions could be used, because the only thing that is required here is that the three equations are independent. The particular choice of paths that is made simply makes the equations simpler.
These three equations in the unknown $v_T,$ $v_N$, $v_W$ and known $v_1$,$v_2$,$v_3$ are of quadratic order, hence writing them in a matrix form is not going to help you solve them. The first step to solve them is finding formulas that expresses $v_N$ and $v_W$ in terms of $v_t$,$v_1$,$v_2$,$v_3$ so that they can be eliminated. This procedure is somewhat similar to eliminating variables in a set of linear equations but more complicated. P,Q,R are not unknown variables, but merely short hand notations in terms $v_1$,$v_2$,$v_3$ to keep the formulas compact. The result is an equation in the unknown $v_T$, the quantity of interest, and the known  values $v_1$,$v_2$,$v_3$ only. It turns out to be a 4th order polynomial equation that can be solved.
The set of three equations has four solutions. There are two types of solutions, one corresponding to $v_T^2>v_N^2+v_W^2$ and one with $v_T^2<v_N^2+v_W^2$, respectively a true velocity larger or smaller than the wind velocity. The other two solutions are obtained from the symmetry where if $(v_T,v_N,v_W)$ is a solution than $(-v_T,-v_N,-v_W)$ is also a solution, which can be seen directly from the original three equations.
The whole problem is already stated in terms of the component of vectors, but using this particular approach will in general result in quadratic equations. It could this be measured differently? Yes of course. If you would fly directly into and with the wind only two flight paths would be required. This is effectively the case with $v_W=0$, but the horseshoe heading is based on an unknown wind velocity and direction. An alternative approach would be to measure other ground properties, i.e., In flying only with north and south heading in combination with measuring both angles between the ground velocity and the heading direction, the equations would be linear.
However, this horseshoe heading technique enables a pilot to determine the three different velocity values from the readily available ground velocities via GPS measurements. This is a more "practical" limitation to the problem.
