requires [[Linear Algebra]] Inspired by https://youtu.be/60z_hpEAtD8 equivalence requires the same magnitude and orientation # Primitives Scalar Vector Magnitude Bivector Trivector k-vector Recall: Inner product $\vec{u}\cdot\vec{v}$, dot product, projection, commutativity, distributivity, and linearity (does *not* capture the *direction* of *rotation* of $\vec{v}$ with respect to $\vec{u}$) Outer product $\vec{u}\land\vec{v}$, bivector, anticommutative, ditributivity, and linearity (does *not* capture the *direction or scale* of *projection* of $\vec{v}$ onto $\vec{u}$) New: Geometric product $\vec{u}\vec{v}=\vec{u}\cdot\vec{v}+\vec{u}\land\vec{v}$, a scalar and a bivector (compensates by including both products) ##### Examples: Squaring a vector: $\vec{u}\vec{u}=\vec{u}\cdot\vec{u}+\vec{u}\land\vec{u}=\|\vec{u}\|^2+0=\|\vec{u}\|^2$ Vector times inverse $\vec{u}\vec{u}^{-1}=\vec{u}\cdot\frac{\vec{u}}{\|\vec{u}\|^2}+\vec{u}\land\frac{\vec{u}}{\|\vec{u}\|^2}=1+0=1$