Vectors may be added, subtracted, multiplied by regular numbers (scalars), and multiplied with each other in two different ways. These operations will now be briefly reviewed.
Addition: Our world seems to be constructed in such a way that vector addition is often the correct thing to do. You are all familiar with the concept of adding two force vectors to obtain the total force. This might seem natural to you now, but it is a miracle that this simple operation describes what actually happens in the world. To add two vectors, we simply add their components, e.g.,
Subtraction: Subtraction is, of course, just another form of addition, but
it is worth mentioning here because we often want
to find the position vector, 
, that
starts at point 1, in three-dimensional space, and ends at point 2.
For instance, the electric force exerted by a charge at point 1 on another
charge at point 2 depends on .
By drawing a picture on which points 1 and 2 are shown and where
is drawn, and by thinking about how you would move along each axis to get
from point 1 to point 2, it is easy to see that the formula for
is given by
where 
is the position vector from the origin to point 1, and
where 
is the position vector from the origin to point 2.
Multiplication by a Scalar: To multiply a vector by a scalar, simply multiply each of its components by the scalar. For instance,
This is often a very handy thing to do; for instance, in the example above we just doubled the velocity of an object. Multiplication by a scalar can also be used to reverse the direction of a vector: simply multiply by -1.
Scalar Product: The simplest way to multiply two vectors is called the scalar product (often called the dot product). The scalar product of two vectors, A and B, is given by
What is this operation good for?
We usually use the scalar product when what is important is not
the whole magnitude of a vector, but only its magnitude along some
direction.
For instance, if a football player is running toward the goal line
at an angle, his total speed is not nearly so important as his
speed toward the goal line.
If we set up a coordinate system in which the 
direction
points from one goal post to the other, and if the football player's
velocity is 
, then the speed at which he is approaching
the goal line is simply 
.
Since our vectors are often written in unit vector form, it is
helpful to memorize the rules for taking the scalar product of
unit vectors:
The scalar product is also useful if we want to know the angle between
two vectors.
Consider the two position vectors 
and 
.
It is easy to compute their scalar product from their components:
To find the angle between them, we use the other scalar product formula
you might remember,
Since we already have the dot product, we only need the magnitudes of
and .
The angle between them is then given by
Vector Product: The vector product (often called the cross product) is more complicated, and is used whenever something twists or circulates. Whenever you unscrew a lid or watch a toilet flush, think cross product. There are several ways to compute the vector product, but the most useful way for this course is to memorize the rules for the cross products of the unit vectors. And the easiest way to memorize these rules is to remember the following diagram:
To take the cross product of two of these unit vectors, find them in the diagram above. If an arrow points from the first vector in the product to the second vector in the product, the answer is simply the third vector in the diagram. If an arrow points from the second vector in the product to the first, then the answer is the negative of the third vector in the diagram. And, best of all, if the two vectors in the product are the same, the answer is zero. Well, this sounds confusing, so here are some examples. Check each one against the rules and the diagram and make sure you understand how to use them; this will come in handy later.
There are nine possible products; six are given here. See if you can work out the other three.