**Vector Notation**: a *vector* quantity is one that has
both *magnitude* and *direction*. Another (equivalent)
way of putting it is that a vector quantity has several *components*
in *orthogonal* (perpendicular) directions.
The idea of a vector is very abstract and general;
one can define useful *vector spaces* of many sorts,
some with an infinite number of orthogonal *basis vectors*,
but the most familiar types are simple 3-dimensional quantities
like position, speed, momentum and so on. The conventional
notation for a vector is ,
sometimes written
or
or **A** but most clearly recognizable when in boldface
with a little arrow over the top. On the blackboard a vector
may be written with a tilde *underneath*, which is hard to
generate in L^{A}TEX.

**Unit Vectors**:
In Cartesian coordinates (*x*,*y*,*z*) a vector
can be
expressed in terms of its three scalar components
*A*_{x},*A*_{y},*A*_{z}
and the corresponding *unit vectors*
(sometimes written as
or occasionally
as
)
thus:

where the little "hat" over a symbol means (in this context) that it has unit magnitude and thus imparts

A unit vector
can be formed from
any vector
by dividing it by its own *magnitude* *a*:

(2) |

Already we have used a bunch of concepts before defining them properly, the usual chicken-egg problem with mathematics. Let's try to catch up:

**Multiplying or Dividing a Vector by a Scalar**:
Multiplying a vector
by a scalar *b* has no effect
on the *direction* of the result (unless *b*=0)
but only on its *magnitude* and/or the *units*
in which it is measured - if *b* is a pure number,
the units stay the same; but multiplying a velocity by a mass
(for instance) produces an entirely new quantity,
in that case the momentum.

Dividing a vector by a scalar *c* is the same as
*multiplying* it by 1/*c*.

This type of product always commutes: .

**Adding or Subtracting Vectors**:
In two dimensions one can draw simple diagrams depicting
"tip-to-tail" or "parallelogram law" vector addition
(or subtraction); this is not so easy in 3 dimensions,

so we fall back on the algebraic method of *adding components*.
Given
from Eq. (1) and

we write

Subtracting from is the same thing as adding .

**Multiplying Two Vectors . . . **

**. . .****to get a Scalar**: we just add together the products of the components,

also known as the "dot product", which commutes: .**. . .****to get a Pseudovector**:

This "cross product" is actually a*pseudovector*(or, more generally, a*tensor*), because (unlike the nice dot product) it has the unsettling property of*not commuting*( ) but we often treat it like just another vector.

We are going to be using these things continually throughout the rest of the course, so make sure you are adept with them.

Jess H. Brewer - Last modified: Fri Nov 13 17:26:28 PST 2015