# 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 0 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: v. We write |v| or ||v|| 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 P1 and P2, then the vector P1P2 has magnitude equal to the distance between the two points and the same direction as a ray that starts at P1 and goes through P2.

In particular, we are often interested in the vector OP, 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 i is the unit vector in the same direction as the positive X axis, the vector j the unit vector in the same direction as the positive Y axis, and the vector k the unit vector in the same direction as the positive Z axis.

### Exercises:

1. What is PP?

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

3. To what points do the vectors i, j, and k correspond?

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

5. Show that PQ = P'Q' if and only if PP' = QQ'.

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

thomasc@athena.mit.edu