Yup. Geometric algebra has one equation for each of: projection, rejection, join (e.g., two points into a line), and meet (e.g., two planes interacting at a lin

Yup. Geometric algebra has one equation for each of: projection, rejection, join (e.g., two points into a line), and meet (e.g., two planes interacting at a line). The algebraic expression is the same regardless of the elements you're working with or the space you're working in. You get transformations too, as easy as M=b/a, where M can be applied to any element in the algebra by taking the square root and applying double-sided multiplication such that b = √M a ~√M, where tilde represents the reverse. These transformations are isomorphic to complex numbers, quaternions, and hypercomplex numbers, and understanding them makes other explanations of these concepts feel inadequate and woefully un-geometric. Add in logarithms and the exponential map for these transformations and we can perform linear interpolation between states and parametrize transformations. I'm just a motivated amateur and I can do all of these things. The vector algebra I learned in engineering is useful, and it's often all I need for simple 3-dimensional problems, but it's just shy of something far more powerful and far more general.