Incidentally, this is how LCDs work: two orthogonal polarizers (so no light should pass through), and sandwiched between them liquid crystals which rotate the l

Incidentally, this is how LCDs work: two orthogonal polarizers (so no light should pass through), and sandwiched between them liquid crystals which rotate the light so it can pass through. When a current is applied, the crystals align, effectively turning off the rotation and thus the light. It's also a nice exercise to see the polarizer as a linear transformation on the horizontal and vertical direction of the light: if the polarizer is oriented at direction θ, it's equivalent to rotating the light by -θ, keeping only the horizontal component, and rotating back. If you work out the matrixes for rotations 0, π/4, and π/2, you can see that multiplying 0 and π/2 gives the zero matrix, while multiplying 0, π/4, and π/2 keeps a nonzero component.