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 \( (\v{xy})^2 = -1 \).

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

  • 2D Rotations in Space
  • Rather than representing an arbitrary vector as a combination of two basis vectors: \(\mathbf{a} = a_x\mathbf{x} + a_y\mathbf{y}\), in this section we will see that we can simplify the algebra by representing an arbitrary vector as a scale and rotation of a single basis vector: \(\mathbf{a} = r_a\mathbf{x}e^{\mathbf{xy}\theta_a}\)