Vector Addition

Definition of Vector Addition

The vector that is formed by the addition of vectors is called the resultant vector, usually denoted as $\stackrel{\rightharpoonup }{r}$. How to add those vectors can vary in terms of if they are given as points or in geometric representation. While the addition can be done with mathematics for points, it is practical to use the parallelogram law when they are geometrically represented.

Vector Addition Formula

Let's say A and B are points in the plane with their coordinates given as $A=(a_1,a_2)$ and $B=(b_1,b_2)$ respectively. Then, the vector addition formula for $\stackrel{\rightharpoonup }{A}+\stackrel{\rightharpoonup }{B}$ can be written as:

$\stackrel{\rightharpoonup }{A}+\stackrel{\rightharpoonup }{B}=(a_1+b_1,a_2+b_2)$

Properties of Vector Addition

• Commutativity: Changing the order of vectors does not change the sum. $A+B=B+A$

• Associativity: Changing the grouping of additions does not change the sum. $A+(B+C)=(A+B)+C$

• Zero element: The addition of a point with zero equals to the point. If the zero element is $O=(0,0)$, then $A+O=A$

• Additive inverse: If a point A is $A=(a_1,a_2)$, then its inverse is $-A=(-a_1,-a_2)$. When these vectors are added, the sum results in zero.

• $A+(-A)=(a_1,a_2)+(-a_1,-a_2)=(a_1-a_1,a_2-a_2)=(0,0)$

Graphical Vector Addition

How can vector addition be performed graphically? Below are the different methods.

Triangle Law of Vector Addition

The triangle law is a vector addition law. It is also known as the head-to-tail method because the heads and tails of the vectors involved are placed on top of each other while trying to find their sum. The figure below shows what the head and tail of a vector look like.

Fig.1 Showing the head and tail of a vector

Let's see how this law is used. Consider the vectors A and B below.

Fig.2 Showing two vectors A and B

To add the two vectors using the head-to-tail method, follow the following procedures.

1. Place the tail of the second vector on the head of the first vector.
2. To find the sum, draw a resultant vector to connect the tail of the first vector to the head of the second vector.

Fig.3 Addition of two vectors

In the figure above, $\stackrel{\rightharpoonup }{A}+\stackrel{\rightharpoonup }{B}=\stackrel{\rightharpoonup }{C}$.

If there is a third vector, you proceed to place the tail of the third vector on the head of the second vector. The resultant vector will be drawn to connect the tail of the first vector to the head of the second vector.

The Parallelogram Law of Vector Addition

According to the parallelogram law, if two vectors can be represented as two adjacent sides from a common vertex and then completed as if they are forming a parallelogram, then the resultant vector can be found from the diagonal of that parallelogram.

To find $\overrightarrow{u}+\overrightarrow{w}$:

1. Place the vectors' tails together

2. Complete the parallelogram by drawing the two parallel sides.

3. After the parallelogram is completed, draw the diagonal starting at the original vectors' vertex as seen in the figure below.

Fig.4 Showing the addition of two vectors

The parallelogram law can also be used when you're given vectors defined as coordinates.

For points $A=(2,3)$ and $B=(1,3)$, the sum can be found using the parallelogram law, seen in Figure 2.

Vector Subtraction

To understand subtraction, it should first be understood what is the negative of a vector. Let's say, there is a vector A. The negative of this vector is defined as -A. The negative of vector A has the same magnitude as Vector A, however, they are in opposite directions.

Fig. 6 The difference between vector A and the negative of vector A

Parallelogram law for Vector Subtraction

To find $\overrightarrow{u}-\overrightarrow{w}$, it should be thought of as $\overrightarrow{u}+(-\overrightarrow{w})$. Keeping this in mind, we end up with the figure below:

Fig.7 Parallelogram law for vector subtraction

Vector Addition Examples

Let's take some examples.

Let's take another example.

Let's take another example.