Though vectors are used in the study of a number of measurements which include displacement, velocity, acceleration, and force, we are limiting ourselves at this point to displacement problems.

1. One Dimensional Vectors: Vectors in one dimension can be illustrated using a simple x axis.

Example:
 
 

2. Vector Addition with one dimensional vectors: The phrase vector addition is simply arithmetic addition in one dimensional analysis of vector quantities.

Example:
 
 

In a situation where one vector is in a forward or positive direction and the other in a backward direction or a negative direction, the result is the addition of a positive and a negative vector respectively. The resultant would be smaller than the vector having the larger magnitude.

In the above example the first vector has a magnitude of 20 km and the second vector has a magnitude of 15 km. Because both were in the same direction the magnitude of the resultant was 35 km. Had the second vector been in the opposite direction, the resultant vector would have had a magnitude of only 5 km.

It should be noted that in simple one dimensional vector analysis forward and reverse (backwards) can be indicated as positive (+) and negative (-) respectively.
 

3. Two Dimensional Vectors: Resolving Vectors into their "x" and "y" components.

Every two dimensional vector can be described in terms of two vectors, an "x component" and a "y component", such that when combined they make up the original single vector. Below are two examples of single vectors being resolved into their x and y components.

Example 1:
 
 

Resolving a vector 1 km, 30 deg. from N 

Example 2:
 
 

Resolving a vector 1km, 60 deg. from N

4. Two Dimensional Vectors: Adding Vectors.

Adding Vectors in two dimensional scenarios (situations) can be misleading, because, unless the vectors are in the same direction, which makes the circumstances that of adding vectors in one dimension, math functions other than simple addition are required. A typical process is explained below involving vector resolution rather than some other approaches such as the use of Law of Sines and the Law of Cosines.

As an example, let's consider the "Vector Addition" of the two vectors described in Section 3 above. There are two vectors. One is 1 km, 30 degrees from N, and the other is 1 km, 60 degrees from N. To "Add" these two vectors they each need to be resolved into their respective X and Y components. This has already been done above, so we can move on to the last step of "Adding" these two vectors. This requires two overall steps.

Step 1: Add up the x components of the vectors getting a sum of the X components, and adding up the y components getting a sum of the Y components.

Step 2: Treat these two sums, the X sum and the Y sum, as the two sides of a right angle triangle. Find the hypotenuse's length by use of the Pythagorean Theorem, and find the angle between the hypotenuse and the X (sum) side of the triangle using the Inverse Tangent Function of the ratio of the opposite side to the adjacent side. Finally deside on what direction is to be expressed based on this newly obtained vector. This newly obtained vector is the actual sum of the two separate vectors after being Added (combined) together. Observe the "Addition" example below.

Example:
 
 

The Addition of the Two Vectors