Definition

In physics we deal with a number of different types of quantities. Some of these quantities are magnitudes, like length, or mass, or duration of time, which can be expressed as non-negative numbers. Others can take on positive or negative values, like electric current or temperature or electric charge, but can still be expressed by a single number. These quantities are called scalars.

However, there are some quantities, such as velocity or force, which are best represented by a non-negative magnitude and a direction. These are called vectors. Vector quantities are usually written with an arrow above them, like so: \(\vec v\).

(Web browsers don't do this very well so I apologize if some of the vectors in this book look weird.)

Vectors are made up of two pieces: a magnitude and a direction. For example, if I say a car is moving at 50 miles per hour due east, I am expressing the car's velocity as a vector, with magnitude "50 miles per hour" and direction "due east". The magnitude of a vector \(\vec v\) is usually specified with absolute-value bars ( |\(\vec v\)| ), although in some cases we will just write the variable without the hat ( v ). Vectors and their magnitudes usually have units just as scalar quantities do.

The direction of a vector can be specified in a number of different ways. We can draw an arrow pointing in the direction of the vector, of course, or we could give the angle that the vector makes with some standard direction ("40 degrees north of east", for example). One way we can represent the direction of a vector is by dividing the vector by its magnitude

$$\hat v={\vec v\over\abs{\vec v}}$$
which is called its unit vector \(\hat v\). A unit vector is so called because it has a length of one unit: (\(|\hat v|=1\)). All vectors which point in the same direction have the same unit vector regardless of how big they are; thus we can think of it as representing its vector's direction.

The basis vectors in
Cartesian coordinates

Certain unit vectors which are particularly important to us are the basis vectors \(\hat x\), \(\hat y\), and \(\hat z\): in Cartesian coordinates, they point in the positive x, y, and z directions respectively. Thus a car with velocity \(\vec v = (5\hat x) {\rm\frac{m}{s}}\) is moving at 5 meters per second in the +x direction. It is common for \(\hat x\) to point right and \(\hat y\) to point up, but it is not necessarily true: always check for a basis such as in the figure to the right.