Addition and Subtraction of Vectors

In these revision notes for Addition and Subtraction of Vectors, we cover the following key points:

• The meaning of equal, opposite and different vectors
• How to add and subtract two or more vectors in a straight line
• The two rules of vectors addition when they are not in a straight line
• To express the difference of vectors as the sum with the inverse
• To add or subtract vectors using coordinates

Addition and Subtraction of Vectors Revision Notes

By definition, two vectors are equal when they have both the same magnitude and direction (when they are parallel).

Two vectors are opposite when they have equal magnitude but opposite direction (when they are antiparallel).

When two vectors are neither equal, nor opposite, they are different. This includes the cases when vectors: 1) have neither equal magnitude nor the same direction, 2) have equal magnitude but different direction, and 3) when they have the same direction but different magnitude.

The resultant vector is a single vector that virtually replaces the actual vectors and gives the same effect in the object or system they act. It is often denoted as V⃗_net (net vector) as well. V⃗ is a generalization. It may represent any vector quantity.

If we add two vectors in a straight line, the result is a new vector whose magnitude is equal to the sum of magnitudes of each single vector and if we subtract them, the result is a new vector whose magnitude is equal to the difference of magnitudes of each single vector.

When two or more vectors do not lie in a straight line, we use two basic rules to find their sum. They are:

The Triangle (tip-to-tail) Rule

Basically, this rule consists on placing the tail of the second vector to the tip of the first one. Even if the vectors are distant, we can use the method of parallel transportation of these two vectors at a place in which we can put them one after another as required. The resultant vector starts at the tail of the first vector and ends to the tip of the second vector.

The Parallelogram (tail-to-tail) Rule

According to this rule, if we put the two vectors a⃗ and b⃗ at the same starting point (tail), the resultant (sum) vector will be the diagonal of the parallelogram formed by these two vectors (we draw other two sides that are in parallel to the given vectors in order to form the parallelogram).

The same rules can be used for the difference of two vectors as well. We must simply upturn the second vector and then use the addition rules.

If we want to add two vectors when the coordinates are given, we can simply add mathematically the respective components. The same is also true for the subtraction of vectors. We can simply subtract the corresponding coordinates to obtain the coordinates of the vector difference.

The mathematical rules for the addition and subtraction of two vectors AB⃗ and CD⃗ are as follows:

and

This rule is valid for the sum of more than two vectors as well. As for the difference between more than two vectors, you must change the sign in the coordinates of the second vector for each pair of vectors that needs to be subtracted and then finding the sum instead of difference.