Vector Geometry

A position vector is a tuple of numbers, for example,

2D                    3D

One of the basic building blocks of algebraic  geometric translation are the algebraic and geometric forms of vector operations.  Vector sums algebraically are just adding like coordinates,

, , 

Geometrically, sums correspond to the position obtained by drawing one vector at the tip of the first as Mathematica shows in the next figure.  (You can use Mathematica to add vectors analytically and draw them geometrically.)

Since we can do the tips-to-tails in either order, we can also view vector addition as the "parallelogram law of vector addition" that says the sum of two vectors lies along the diagonal of a parallelogram with the vectors on the edges.  This is shown next in 3D.

All the basic linear geometric notions of lengths, angles, areas, volumes, etc. are treated in the chapter on vectors and then used systematically throughout the course together with the local linearization discussed above.  Local linearization lets us extend those computations to nonlinear formulas.

An Interesting Vector Sum

  Click the link below the graph to see this in motion.

Load the Animation