Day 54 in MAT229

Last time I introduced vectors as quantities having both size (length, magnitude) and direction.
Notice that they don't have location: you can move them around in space, and they're the same vector! So you have to place them in some context (a coordinate system).
For vectors in 3D, length (magnitude) was measured by an extension of the Euclidean distance formula between two points:
\[ length=\sqrt{(x_1-x_0)^2+(y_1-y_0)^2+(z_1-z_0)^2} \] where $(x_0,y_0,z_0)$ and $(x_1,y_1,z_1)$ are the coordinates of the two points in Cartesian coordinates where the head and the tail of the vector occur.
So this formula is relative to a coordinate system, whereas the length of the vector is, in fact, independent of coordinate system.
If a vector is in two-dimensions, then the length is just given using the old Euclidean distance of the Pythagorean theorem formula: \[ length=\sqrt{(x_1-x_0)^2+(y_1-y_0)^2} \]
Obviously this can be generalized to vectors in four dimensions, five dimensions; even six dimensions. Maybe seven dimensions, too; probably eight, but certain not not nine dimensions.
Maybe this can be generalized even to n dimensions! (This is especially important for you statisticians).
The equation for a sphere centered at the point $(x_0,y_0,z_0)$ of radius $r$ is given as the set of points equidistant (distance $r$) from the point $(x_0,y_0,z_0)$; that is, the sphere is the set of points $(x,y,z)$ such that
\[ \sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}=r \] or \[ (x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2 \]
This is the set of points P at a distance of r from the point $(x_0,y_0,z_0)$.
Let's remind ourselves of spheres, from Sect. 12.1: 3D coordinate systems (pdf, pdf summary)
We ended class with the concept of unit vectors, which are special vectors of unit length (length 1), and which are building blocks (a basis) for other vectors. They help us to create a coordinate system in which to write vectors.