Vectors

Vectors are quantities that require a direction to be specified in order for the quantity to have full meaning.

Vectors represent magnitudes measured from an origin in some set of coordinate axes.  In standard Cartesian coordinates, a point defined in 2 or 3 dimensional space defines a vector measured from the origin (e.g. (2, -3) or (2, 8, -3.5)).

Analogy of coordinates:

Follow the instructions below:

1.  Take 4 steps North.

2.  Take 3 steps West.

3.  Take 5 steps South.

Now...where are you?  A full answer to this question will be a great example of a displacement vector; you are 3 steps west and 1 step south from your original position.  In other words, you are 3.16 steps at 18 degrees South of West from your original position.

We can describe this position in Cartesian coordinates as (-3,-1) or (-3,-1,0) in 3 dimensions assuming North/South to be the y-axis, and East/West to be the x-axis.

Vector Sum

Vectors can be added together (remember that subtraction is just addition of a negative number)

A vector sum is found graphically by arranging the vectors (graphically represented by arrows) such that each vector arrowhead joins with the back end of the next vector arrow.

If vectors A and B are added together to form vector C, the relationship looks like simple arithmetic:

A + B = C

However, A and A are 2 quite different things; A is a scalar without a direction whereas A is bolded which represents a vector (a vector is usually represented with an arrow over the letter in hand written documents).  This means that if the lengths of the vectors A and B were 4m and 6m respectfully, the direction portion can make the sum look very different as shown below:

Similar diagrams can be created for vector differences as well.  Remember that C - B is simply C + ^-B.  Also, ^-B is simply the magnitude of B, but in the opposite direction of B.  So, C - B is represented as follows: