The original question was:
I got quite the challenge from my father in law. The problem is well defined, but I’m having difficulties finding a meaningful answer. The reason why he asked me is because I’m an engineering student and he is in the windmill industry.
Before they attach the actual mill on the concrete foundation, it has to be absolutely leveled. If not, a tall mill would be quite offset even with a very small angle. To tackle this, they use two angle gauges and measure in two directions. The angle gauges are connected and you know the angle between them, their mutual angle. I’m supposed to find a way to convert these 3 inputs (angle 1, angle 2 and mutual angle) to find 2 outputs (the steepest angle and in which direction this is relative to the gauges).
Physicist: This is a gorgeous question that leads through some pretty math and ends with an elegant answer. If you’ve taken a class or two that used lots of vectors, then this is a cute exploration of what you can do with surprisingly little. If you’ve never taken a class or two that used lots of vectors, then please do: it’s fun stuff. You get to draw pictures and everything.
So you’ve got a flat slab that isn’t quite level. Two angle gauges (with plumb lines or bubbles or whatever) are placed on the slab in two directions. Define 
An angle gauge or “inclinometer”.
Define the angle between 
Measuring the angles between these vectors means that we know sine and cosine of these angles, and knowing that means that we know the dot product and magnitude of the cross product, since 
Finally, since the windmill will be built perpendicular to the slab, it will be built perpendicular to both 
A pristine windmill, slanted, and polluted with a mess of vectors.
If we project 
The questions (way back at the top of this page) now boil down to:
1) What is the angle between
2) What is the angle between
For #1, it turns out that the cross product is easier to work with. Define 
![\begin{array}{ll} |\vec{w}\times\vec{u}|^2 \\[2mm] = |(\vec{a}\times\vec{b})\times\vec{u}|^2 \\[2mm] = |(\vec{u}\cdot\vec{a})\vec{b} - (\vec{u}\cdot\vec{b})\vec{a}|^2 & \textrm{(nobody can remember this identity)} \\[2mm] = (\vec{u}\cdot\vec{a})^2|\vec{b}|^2 + (\vec{u}\cdot\vec{b})^2|\vec{a}|^2 - 2(\vec{u}\cdot\vec{a})(\vec{u}\cdot\vec{b})(\vec{a}\cdot\vec{b}) \\[2mm] = (\vec{u}\cdot\vec{a})^2 + (\vec{u}\cdot\vec{b})^2 - 2(\vec{u}\cdot\vec{a})(\vec{u}\cdot\vec{b})(\vec{a}\cdot\vec{b}) \\[2mm] = \cos^2{(\theta_a)} + \cos^2{(\theta_b)} - 2\cos{(\theta_a)}\cos{(\theta_b)}\cos{(\phi)} \\[2mm] \end{array}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D++%7C%5Cvec%7Bw%7D%5Ctimes%5Cvec%7Bu%7D%7C%5E2+%5C%5C%5B2mm%5D++%3D+%7C%28%5Cvec%7Ba%7D%5Ctimes%5Cvec%7Bb%7D%29%5Ctimes%5Cvec%7Bu%7D%7C%5E2+%5C%5C%5B2mm%5D++%3D+%7C%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Ba%7D%29%5Cvec%7Bb%7D+-+%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bb%7D%29%5Cvec%7Ba%7D%7C%5E2+%26+%5Ctextrm%7B%28nobody+can+remember+this+identity%29%7D+%5C%5C%5B2mm%5D++%3D+%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Ba%7D%29%5E2%7C%5Cvec%7Bb%7D%7C%5E2+%2B+%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bb%7D%29%5E2%7C%5Cvec%7Ba%7D%7C%5E2+-+2%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Ba%7D%29%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bb%7D%29%28%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bb%7D%29+%5C%5C%5B2mm%5D++%3D+%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Ba%7D%29%5E2+%2B+%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bb%7D%29%5E2+-+2%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Ba%7D%29%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bb%7D%29%28%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bb%7D%29+%5C%5C%5B2mm%5D++%3D+%5Ccos%5E2%7B%28%5Ctheta_a%29%7D+%2B+%5Ccos%5E2%7B%28%5Ctheta_b%29%7D+-+2%5Ccos%7B%28%5Ctheta_a%29%7D%5Ccos%7B%28%5Ctheta_b%29%7D%5Ccos%7B%28%5Cphi%29%7D+%5C%5C%5B2mm%5D++%5Cend%7Barray%7D&bg=ffffff&fg=000&s=0 "\begin{array}{ll} |\vec{w}\times\vec{u}|^2 \\[2mm] = |(\vec{a}\times\vec{b})\times\vec{u}|^2 \\[2mm] = |(\vec{u}\cdot\vec{a})\vec{b} - (\vec{u}\cdot\vec{b})\vec{a}|^2 & \textrm{(nobody can remember this identity)} \\[2mm] = (\vec{u}\cdot\vec{a})^2|\vec{b}|^2 + (\vec{u}\cdot\vec{b})^2|\vec{a}|^2 - 2(\vec{u}\cdot\vec{a})(\vec{u}\cdot\vec{b})(\vec{a}\cdot\vec{b}) \\[2mm] = (\vec{u}\cdot\vec{a})^2 + (\vec{u}\cdot\vec{b})^2 - 2(\vec{u}\cdot\vec{a})(\vec{u}\cdot\vec{b})(\vec{a}\cdot\vec{b}) \\[2mm] = \cos^2{(\theta_a)} + \cos^2{(\theta_b)} - 2\cos{(\theta_a)}\cos{(\theta_b)}\cos{(\phi)} \\[2mm] \end{array}")
We also know that:
![\begin{array}{ll} |\vec{w}\times\vec{u}|^2 \\[2mm] = |\vec{w}|^2|\vec{u}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{w}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{a}\times\vec{b}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{a}|^2|\vec{b}|^2\sin^2{(\phi)}\sin^2{(\Omega)} \\[2mm] = \sin^2{(\phi)}\sin^2{(\Omega)} \\[2mm] \end{array}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D++%7C%5Cvec%7Bw%7D%5Ctimes%5Cvec%7Bu%7D%7C%5E2+%5C%5C%5B2mm%5D++%3D+%7C%5Cvec%7Bw%7D%7C%5E2%7C%5Cvec%7Bu%7D%7C%5E2%5Csin%5E2%7B%28%5COmega%29%7D+%5C%5C%5B2mm%5D++%3D+%7C%5Cvec%7Bw%7D%7C%5E2%5Csin%5E2%7B%28%5COmega%29%7D+%5C%5C%5B2mm%5D++%3D+%7C%5Cvec%7Ba%7D%5Ctimes%5Cvec%7Bb%7D%7C%5E2%5Csin%5E2%7B%28%5COmega%29%7D+%5C%5C%5B2mm%5D++%3D+%7C%5Cvec%7Ba%7D%7C%5E2%7C%5Cvec%7Bb%7D%7C%5E2%5Csin%5E2%7B%28%5Cphi%29%7D%5Csin%5E2%7B%28%5COmega%29%7D+%5C%5C%5B2mm%5D++%3D+%5Csin%5E2%7B%28%5Cphi%29%7D%5Csin%5E2%7B%28%5COmega%29%7D+%5C%5C%5B2mm%5D++%5Cend%7Barray%7D&bg=ffffff&fg=000&s=0 "\begin{array}{ll} |\vec{w}\times\vec{u}|^2 \\[2mm] = |\vec{w}|^2|\vec{u}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{w}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{a}\times\vec{b}|^2\sin^2{(\Omega)} \\[2mm] = |\vec{a}|^2|\vec{b}|^2\sin^2{(\phi)}\sin^2{(\Omega)} \\[2mm] = \sin^2{(\phi)}\sin^2{(\Omega)} \\[2mm] \end{array}")
And therefore:
In the event that 
For #2 we find the projection, 
![\begin{array}{ll} \vec{a}\cdot\vec{p} \\[2mm] = \vec{a}\cdot\left(\vec{u} - \frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}\right)\\[2mm]=\vec{a}\cdot\vec{u}-\frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{a}\cdot\vec{w}\\[2mm]=\vec{a}\cdot\vec{u} \end{array}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D++%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bp%7D+%5C%5C%5B2mm%5D++%3D+%5Cvec%7Ba%7D%5Ccdot%5Cleft%28%5Cvec%7Bu%7D+-+%5Cfrac%7B%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%7D%7B%5Cvec%7Bw%7D%5Ccdot%5Cvec%7Bw%7D%7D%5Cvec%7Bw%7D%5Cright%29%5C%5C%5B2mm%5D%3D%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bu%7D-%5Cfrac%7B%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%7D%7B%5Cvec%7Bw%7D%5Ccdot%5Cvec%7Bw%7D%7D%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bw%7D%5C%5C%5B2mm%5D%3D%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bu%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000&s=0 "\begin{array}{ll} \vec{a}\cdot\vec{p} \\[2mm] = \vec{a}\cdot\left(\vec{u} - \frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}\right)\\[2mm]=\vec{a}\cdot\vec{u}-\frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{a}\cdot\vec{w}\\[2mm]=\vec{a}\cdot\vec{u} \end{array}")
In that last step you know that 
Defining 
![\begin{array}{ll} (\vec{a}\cdot\vec{p})^2 = |\vec{a}|^2|\vec{p}|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow (\vec{a}\cdot\vec{u})^2 = |\vec{p}|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow \cos^2{\theta_a}=\left|\vec{u} - \frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\vec{w}\right|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(|\vec{u}|^2 + \left(\frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\right)^2|\vec{w}|^2 - 2\frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\vec{u}\cdot\vec{w}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(1 + \frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2} - 2\frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(1-\frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(1 - \frac{|\vec{u}|^2|\vec{w}|^2}{|\vec{w}|^2}\cos^2{(\Omega)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(1 - \cos^2{(\Omega)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\sin^2{(\Omega)} \end{array}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D++%28%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bp%7D%29%5E2+%3D+%7C%5Cvec%7Ba%7D%7C%5E2%7C%5Cvec%7Bp%7D%7C%5E2%5Ccos%5E2%7B%28%5Comega_a%29%7D+%5C%5C%5B2mm%5D++%5CRightarrow+%28%5Cvec%7Ba%7D%5Ccdot%5Cvec%7Bu%7D%29%5E2+%3D+%7C%5Cvec%7Bp%7D%7C%5E2%5Ccos%5E2%7B%28%5Comega_a%29%7D+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D%3D%5Cleft%7C%5Cvec%7Bu%7D+-+%5Cfrac%7B%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%7D%7B%7C%5Cvec%7Bw%7D%7C%5E2%7D%5Cvec%7Bw%7D%5Cright%7C%5E2%5Ccos%5E2%7B%28%5Comega_a%29%7D+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D%3D%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Cleft%28%7C%5Cvec%7Bu%7D%7C%5E2+%2B+%5Cleft%28%5Cfrac%7B%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%7D%7B%7C%5Cvec%7Bw%7D%7C%5E2%7D%5Cright%29%5E2%7C%5Cvec%7Bw%7D%7C%5E2+-+2%5Cfrac%7B%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%7D%7B%7C%5Cvec%7Bw%7D%7C%5E2%7D%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%5Cright%29+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D%3D%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Cleft%281+%2B+%5Cfrac%7B%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%29%5E2%7D%7B%7C%5Cvec%7Bw%7D%7C%5E2%7D+-+2%5Cfrac%7B%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%29%5E2%7D%7B%7C%5Cvec%7Bw%7D%7C%5E2%7D%5Cright%29+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D%3D%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Cleft%281-%5Cfrac%7B%28%5Cvec%7Bu%7D%5Ccdot%5Cvec%7Bw%7D%29%5E2%7D%7B%7C%5Cvec%7Bw%7D%7C%5E2%7D%5Cright%29+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D+%3D+%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Cleft%281+-+%5Cfrac%7B%7C%5Cvec%7Bu%7D%7C%5E2%7C%5Cvec%7Bw%7D%7C%5E2%7D%7B%7C%5Cvec%7Bw%7D%7C%5E2%7D%5Ccos%5E2%7B%28%5COmega%29%7D%5Cright%29+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D+%3D+%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Cleft%281+-+%5Ccos%5E2%7B%28%5COmega%29%7D%5Cright%29+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D+%3D+%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Csin%5E2%7B%28%5COmega%29%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000&s=0 "\begin{array}{ll} (\vec{a}\cdot\vec{p})^2 = |\vec{a}|^2|\vec{p}|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow (\vec{a}\cdot\vec{u})^2 = |\vec{p}|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow \cos^2{\theta_a}=\left|\vec{u} - \frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\vec{w}\right|^2\cos^2{(\omega_a)} \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(|\vec{u}|^2 + \left(\frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\right)^2|\vec{w}|^2 - 2\frac{\vec{u}\cdot\vec{w}}{|\vec{w}|^2}\vec{u}\cdot\vec{w}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(1 + \frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2} - 2\frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}=\cos^2{(\omega_a)}\left(1-\frac{(\vec{u}\cdot\vec{w})^2}{|\vec{w}|^2}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(1 - \frac{|\vec{u}|^2|\vec{w}|^2}{|\vec{w}|^2}\cos^2{(\Omega)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(1 - \cos^2{(\Omega)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\sin^2{(\Omega)} \end{array}")
Again, in the event that 
![\begin{array}{ll} \cos^2{\theta_a} = \cos^2{(\omega_a)}\sin^2{(\Omega)} \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(\cos^2{(\theta_a)} + \cos^2{(\theta_b)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}\left(1-\cos^2{(\omega_a)}\right) = \cos^2{(\omega_a)}\cos^2{(\theta_b)} \\[2mm] \Rightarrow \cos^2{\theta_a}\sin^2{(\omega_a)} = \cos^2{(\omega_a)}\cos^2{(\theta_b)} \\[2mm] \Rightarrow \tan^2{(\omega_a)} = \frac{\cos^2{(\theta_b)} }{\cos^2{(\theta_a)}} \\[2mm] \Rightarrow \left|\tan{(\omega_a)}\right| = \left|\frac{\cos{(\theta_b)} }{\cos{(\theta_a)}}\right| \\[2mm] \end{array}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D++%5Ccos%5E2%7B%5Ctheta_a%7D+%3D+%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Csin%5E2%7B%28%5COmega%29%7D+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D+%3D+%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Cleft%28%5Ccos%5E2%7B%28%5Ctheta_a%29%7D+%2B+%5Ccos%5E2%7B%28%5Ctheta_b%29%7D%5Cright%29+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D%5Cleft%281-%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Cright%29+%3D+%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Ccos%5E2%7B%28%5Ctheta_b%29%7D+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ccos%5E2%7B%5Ctheta_a%7D%5Csin%5E2%7B%28%5Comega_a%29%7D+%3D+%5Ccos%5E2%7B%28%5Comega_a%29%7D%5Ccos%5E2%7B%28%5Ctheta_b%29%7D+%5C%5C%5B2mm%5D++%5CRightarrow+%5Ctan%5E2%7B%28%5Comega_a%29%7D+%3D+%5Cfrac%7B%5Ccos%5E2%7B%28%5Ctheta_b%29%7D+%7D%7B%5Ccos%5E2%7B%28%5Ctheta_a%29%7D%7D+%5C%5C%5B2mm%5D++++%5CRightarrow+%5Cleft%7C%5Ctan%7B%28%5Comega_a%29%7D%5Cright%7C+%3D+%5Cleft%7C%5Cfrac%7B%5Ccos%7B%28%5Ctheta_b%29%7D+%7D%7B%5Ccos%7B%28%5Ctheta_a%29%7D%7D%5Cright%7C+%5C%5C%5B2mm%5D++%5Cend%7Barray%7D&bg=ffffff&fg=000&s=0 "\begin{array}{ll} \cos^2{\theta_a} = \cos^2{(\omega_a)}\sin^2{(\Omega)} \\[2mm] \Rightarrow \cos^2{\theta_a} = \cos^2{(\omega_a)}\left(\cos^2{(\theta_a)} + \cos^2{(\theta_b)}\right) \\[2mm] \Rightarrow \cos^2{\theta_a}\left(1-\cos^2{(\omega_a)}\right) = \cos^2{(\omega_a)}\cos^2{(\theta_b)} \\[2mm] \Rightarrow \cos^2{\theta_a}\sin^2{(\omega_a)} = \cos^2{(\omega_a)}\cos^2{(\theta_b)} \\[2mm] \Rightarrow \tan^2{(\omega_a)} = \frac{\cos^2{(\theta_b)} }{\cos^2{(\theta_a)}} \\[2mm] \Rightarrow \left|\tan{(\omega_a)}\right| = \left|\frac{\cos{(\theta_b)} }{\cos{(\theta_a)}}\right| \\[2mm] \end{array}")
Similarly, 
So, if you’ve got the inclinometer readings, 
The windmill picture is from here.
4 Responses to Q: How do I know my windmill is on straight?