What is the significance of cos²α+cos²β+cos²γ = 1 in crystallography?

[RK7]

On pg 6 of http://www.scribd.com/doc/3914281/Crystal-Structure, it quotes this result without proof. My notes from uni also quote this result but I can't see where it comes from. Does anyone know? Thanks

 

[mathman]

It looks like the Pythagorean theorem applied in three dimensions.

 

[HallsofIvy]

Well, it isn't true for just any angle \alpha, \beta, \gamma, of course! If, however, you draw a line through the origin of a three dimensional coordinate system, define \alpha to be the angle the line makes with the x-axis, \beta to be the angle the line makes with the y-axis, and \gamma to be the angle the line makes with the z-axis, then this is true.

To see that, think of a vector of unit length in the direction of that line. If we drop a perpendicular from the tip of the vector to the x-axis, we have a right triangle in which an angle is \alpha and the hypotenuse is 1, the length of the vector. Thus, the projection of the vector on the x-axis, and so the x-component of the vector is cos(\alpha). Similarly, the y-component of the vector is cos(\beta) and the z-component of the vector is cos(\gamma). That is, cos^2(\alpha)+ cos^2(\beta)+ cos^2(\gamma) is the square of the length of the vector which is, of course, 1.

By the way, look what happens if you do this in two dimensions. If you have a line in the plane through the origin making angles \alpha with the x-axis and \beta with the y- axis then \beta= \pi/2- \alpha so cos^2(\alpha)+ cos^2(\beta)= cos^2(\alpha)+ cos^2(\pi/2- \alpha)= cos^2(\alpha)+ sin^2(\alpha)= 1.

 

[RK7]

That was embarrassing.. thanks