Rotations in Space

At the end of the previous section we saw that we can rotate an arbitrary vector by \(\frac{\pi}{2}\) 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, states that for any real number \(x\),

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

    To appreciate Euler's formula and it's application to geometric algebra, and ultimately relativity, we need to understand why the imaginary exponent of this special number \(e\) results in the trigonometric functions.

    If you already understand how to derive Euler's formula, you may want to skip over to Euler's formula and Geometric Algebra.

  • 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 \( (\mathbf{xy})^2 = -1 \).

    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: \(\mathbf{a} = r_a\mathbf{x}e^{\mathbf{xy}\theta_a}\)