# 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: $$\mathbf{a} = r_a\mathbf{x}e^{\mathbf{xy}\theta_a}$$