# Vectors

Pre-requisites: Cartesian Coordinates
A **vector** is the combination of a **magnitude** and a **direction**.
The magnitude or size of a vector is just a positive real number, and
direction can be defined mathematically as a ray with its base point at the
origin. We also allow the zero vector
which has 0 magnitude and no direction, and always distinguish vector
variables from other real number or **scalar** variables by putting a
little vector hat over them like so:
. We write
to denote the magnitude
of a vector, and call any vector with magnitude 1 a **unit vector**.

Physics provided numerous examples of vectors. For example, force is
a vector quantity: you can't apply a force without applying it in some
direction, and every force has a magnitude corresponding to its strength.
Other examples of vector quantities include the velocity and acceleration
of a particle, the angular momentum of a rigid body, and torque.

One important instance of vectors in mathematics is the displacement
between two points. I have two points *P*_{1} and
*P*_{2}, then
the vector
has magnitude equal to the distance between the two points and the same
direction as a ray that starts at *P*_{1}
and goes through *P*_{2}.

In particular, we are often interested in the vector
, the displacement
between the point *P* and the origin. Every point generates
such a vector, and furthermore, every vector can be regarded as the
displacement between the origin and some point. Because of this
correspondence, we often blur the distinction between points and
vectors.
This blurring can become confusing when we introduce cartesian coordinates,
so be careful. The tuple (2,3.5,-7) can mean either the point with
those coordinates, or the vector giving the displacement between the
origin and that point. Only the surrounding context can determine which
is meant. Introducing cartesian coordinates does give us three new
vectors to play around with, however: the vector
is the unit vector in
the same direction as the positive X axis, the vector
the unit vector in
the same direction as the positive Y axis, and the vector
the unit vector in
the same direction as the positive Z axis.

### Exercises:

- What is
?

- What is the magnitude of the vector which represents the displacement
between the point (1,2,3) and the point (4,5,6)?

- To what points do the vectors
correspond?

- Describe the vector corresponding to the force that gravity exerts
on a person near the surface of the Earth.

- Show that
.

Solutions to Exercises |
Vector Calculus Index |
World Web Math Main Page

thomasc@athena.mit.edu
Last Modified 1 July 1997