Math and science were invented by humans to describe and
understand the world around us.
We observe that there are some quantities and processes in
our world that depend on the **direction** in which
they occur, and there are some quantities that do not depend
on direction.
Mathematicians and scientists call a quantity
which depends on direction a **vector quantity**. A quantity
which does not depend on direction is called a **scalar quantity**.
A
vector quantity
has two characteristics, a **magnitude** and a **direction**.
When comparing
two vector quantities of the same type, you have to compare both
the magnitude and the direction.

On this slide we show three examples in which two vectors are being
compared.
Vectors are usually denoted on figures by an arrow.
The length of the arrow indicates the magnitude and the
tip of the arrow indicates the direction. The vector is
labeled with an alphabetical
letter with a line over the top to distinguish it from a scalar.
Our web print fonts don't allow for this notation, so we will use
a bold letter for a vector.
We will be comparing two vectors, **a**
and **b**. They could be forces, or velocities, or accelerations;
it doesn't really matter.

Example #1: We have two vectors with the same direction, but the
magnitudes (or length of the vectors) are different. Vector **a**
does not equal vector **b** in this example. This example seems
pretty simple, because the same rule applies for scalars; if the
magnitude is different, the quantities are not equal. An object
weighing 50 pounds is not equal to an object weighing 25 pounds.

Example #2: This example is a little more complex.
In this case, we have two vectors with
equal magnitude, but the directions are different. Vector **a**
does not equal vector **b** in this example. If the vector was a
velocity, this tells us that a car traveling 45 mph to the northeast
will end up in a different place than another car also traveling 45 mph
due east. In one hour, they will both move 45 miles, but the locations
will be different. In two hours, they will be even farther apart.

Example #3: In this example, we have two vectors with equal length
and equal direction. Vector
is equal to vector **b**.
For two vectors to be equal, they must have both the magnitude and the
directions equal.

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