While complex numbers define a point, vectors define a line. This is important to remember because now you have to keep in mind both the magnitude and the direction of the line when dealing with vectors. Vectors start at one point and end at another point. It is convenient to describe vectors that start at the origin (denoted O) and end at a point (denoted P). The point that has the arrow defines the direction.
Vectors can be denoted as 
or 
Typically, vectors are denoted simply as a single variable, such as 
or A.
Vectors that start at the origin and end at another point can be described simply as coordinates. This is known as the rectangular form. An example is 4i + 3j. The “i” in vectors is not the same “i” from complex numbers, but vectors in this form are graphed in the same way. In this case, the i part is a unit vector in the positive x direction, and the j part is a unit vector in the positive y direction. The origin of the axes also serves as the origin for the vector. You may also see them defined as a magnitude and an angle only, like 5, 45°. This is known as the polar form and tells you the magnitude (or length) of the vector is 5 units at an angle of 45°. Get used to seeing vectors in both the rectangular form and the polar form because both of these forms are prevalent.
Similar to complex numbers, you can perform mathematical operations on vectors.
The sum or difference of vectors is called a resultant because it’s the vector that is the result of adding or subtracting vectors. To be able to find the resultant:
Example:
Find the resultant 
in both rectangular and polar form, of the following two vectors: 
and 
given that 
and that sin 60° = 0.87.
Solution:
Ax + Bx = 2.61 + 2.5 = 5.11
Ay + By = 1.5 + 4.35 = 5.85
In rectangular form, the resultant is 
In polar form, the resultant is 
Vector multiplication has many meanings. Conceptually, it is not very straightforward, and you need to understand what the vectors represent to understand what type of multiplication you will want to use. The two most common types of multiplication of vectors are briefly covered below.
Dot Product: Let 
and 
be vectors and let θ be the angle between these vectors. The dot product of the vectors 
and 
is found using the following equation:
Because the dot product results in a number, or a magnitude, it is often referred
to as the scalar product. Let 
be a nonzero vector. Then 
is a unit vector in the direction of 
The dot product of 
and 
is the magnitude of the component of the vector 
In the direction of the vector 
Therefore:
Example:
Find the dot product of the two vectors 
and 
from the previous example.
Solution:
Cross Product: The cross product allows you to find a new vector that is perpendicular to the plane of the two vectors being multiplied. Because you are finding a vector, this is also referred to as the vector product. Use the following equation to find the cross product:
Here, θ represents the smaller angle between the two vectors and n is a unit vector perpendicular to the plane formed by the vectors 
and 
The magnitude of 
is 
sin θ. The direction of 
is determined by the right hand rule. The right hand rule says that when you rotate
the right hand from 
to 
through the smaller angle between 
and 
then your thumb points in the direction of 
This is the direction of the unit vector n.
Example:
Given the vectors 
and 
from the previous two examples, find the cross product.
Solution:
The magnitude of 
is 7.5. Here, n represents the unit vector in the positive z direction. Remember, 
is a vector in the plane perpendicular to the plane of vectors 
and 
The right hand rule says that 
is in the positive z direction. So 
is 7.5 units long in the positive z direction.