1. Vectors

Vectors have length and direction. Often, we'll be using a vector v to indicate a direction, and will talk of ``the direction v''. The length or norm or magnitude of v is $\vert$v$\vert$ or $\vert$   v$\vert$.

If u has length 1, it is a unit vector. If v is any nonzero vector, then u$= \frac {\mbox{\bf v}} {\vert\mbox{\bf v}\vert}$ is a unit vector with the same direction as v, and u is said to be obtained by normalizing v. Here the word ``normal'' means ``of length 1''.

Be careful--the word ``normal'' is also used to mean ``perpendicular''.

A unit vector is often used to describe a direction. Its components are sometimes called its direction cosines, for a reason explained under C.

Addition and subtraction of vectors are performed ``coordinatewise''. So is multiplication of a vector by a scalar.

Handy: If $P$ and $Q$ are points, the vector from $P$ to $Q$ is $Q-P$.