Vector Definitions

Vector Representations

We define a vector as an arrow in space. We can begin to look at vectors on the xy-plane, but ultimately we will consider vectors in 3-space.At right we have drawn a vector on the xy-plane. The name of a vector is usually written in bold type when printed but for hand-written vectors, we usually use an over-arrow like this.

Sometimes vectors are specified by naming their endpoints. The vector that starts at point A and ends at point B will be written as

We can describe the vector exactly by describing it horizontal extension and its vertical extension. We always go from the beginning of the arrow to the end point. Then these two components are written

as shown in the drawing.

The vector has an angle of inclination that it makes with the positive x-direction which we can call

. Therefore

The length of the vector, known as its magnitude, is specified as

It should also be clear from the diagram that

Pure numbers are referred to as scalars to distinguish them from vectors.

We can multiply a vector by a scalar. This changes its length but not its angle of inclination:

and

on the xy-plane.

Keep in mind that all vectors are considered to be "free".

There is no difference between the vectors shown in the diagram at left.

They are all the same vector.

Two vectors are said to be parallel if one is a multiple of the other. They will point in the same direction if one is a positive multiple

of the other and the opposite direction if one is a negative multiple of another.

In this diagram, and making all three vectors parallel since they are all three scalar multiples of one another.

Vector Addition

Two vectors are added by adding their components.

If and then .

We call the new vector

the resultant vector.

In the diagram at left you can see what happens when is added to .Geometrically, one vector is placed at the end of the other vector. This can be done as either