 # The Geometric Algebra of 3D Euclidean Space The Rules of the Game Let us start with vectors. A vector, is a directed number. A number consisting of two parts: an amount, and a direction. It can be visualized as an arrow (see Figure 1.1). The length of the arrow represents the amount and the direction is just the direction of the arrow. Vectors are pieces of oriented lines. There is a sense of going forward on the line and backwards on the line. Parallel arrows of the same length and with the same orientation are represented by the same vector no matter where the arrow is located. This is why the algebra of vectors is so useful for describing displacements, and forces and many other physical quantities. Similarly with bivectors. Bivectors are directed numbers in two dimensions. They also have two parts an amount and a direction. The amount is a two-dimensional amount (an area) and the direction is a two-dimensional direction the direction of a plane. Vectors are pieces of oriented lines, bivectors are pieces of oriented planes. There is a sense of going forward (covering more area) on the plane and a sense of going backwards (covering less area) on the plane. You may think of it as unrolling a roll of cloth or rolling it back. Two pieces with the same area on two parallel planes with the same orientation are represented by the same bivector no matter what the shape of the area looks like or where the planes are located. The unit vectors e1 and e2 are orthogonal. The angle measured from e1 to e2 is 90 degrees (usually measured counter-clockwise). The multiplication by (e1 e2) on the right of a vector rotates the vector 90 degrees in the same direction that e1 goes towards e2 (usually counter-clockwise). Let us now see what happens when we post-multiply by the product of two unit vectors u v that are now separated by an arbitrary angle t. In other words let u and v be vectors with u^2 = v^2 = 1