Rotations in Space

At the end of the previous section we saw that we can rotate an arbitrary vector by 90 degrees, \(\frac{\pi}{2}\) radians, simply by multiplying that vector by \(\mathbf{xy}\). In this section we will see that this is because \(\mathbf{xy} = e^{\mathbf{xy}\frac{\pi}{2}}\) and that more generally we can rotate the same vector by any arbitrary angle \(\theta\) simply by multiplying by \(e^{\mathbf{xy}\theta}\).

But to get to that point we'll first need to take a closer look at Euler's formula and its application to geometric algebra - where we no longer need to define the imaginary unit \(i\).

  • Euler's Formula
  • Euler's formula, the most remarkable formula in mathematics, according to Richard Feynman and many others, states that for any real number \(x\),

    \[ e^{ix} = \cos x + i \sin x \]

    In this section we will derive Euler's formula to get a better understanding of how Euler's formula can be used to describe rotations in space and time, eventually including the Lorentz transformation.

  • Euler's Formula and Geometric Algebra
  • While working to understand and derive Euler's formula we introduced an imaginary unit \(i\) with the property that \(i^2 = -1\). But we've already seen that the product of the two basis vectors has this same property in that \( (\v{xy})^2 = -1 \), though it differs in other properties as we shall see.

    In this section we will investigate the properties of various rotations in 2D planes using Euler's formula and Geometric Algebra in two dimensional space.