We may not realize it but we do a bit of addition every day. We do it when we go to the grocery store to buy some items, we do it when adding ingredients to our food while cooking, and even while playing with friends. It is part of our everyday life and it can also be applied to things a little bit more complex like vectors.

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Vector addition can be defined as the procedure of adding two or more vectors.

The vector that is formed by the addition of vectors is called the resultant vector, usually denoted as $\stackrel{⇀}{r}$. 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.

Let's say A and B are points in the plane with their coordinates given as $A=\left({a}_{1},{a}_{2}\right)$ and $B=\left({b}_{1},{b}_{2}\right)$ respectively. Then, the vector addition formula for $\stackrel{⇀}{A}+\stackrel{⇀}{B}$ can be written as:

$\stackrel{⇀}{A}+\stackrel{⇀}{B}=\left({a}_{1}+{b}_{1},{a}_{2}+{b}_{2}\right)$

• 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. $A+\left(B+C\right)=\left(A+B\right)+C$

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

• Additive inverse: If a point A is $A=\left({a}_{1},{a}_{2}\right)$, then its inverse is $-A=\left(-{a}_{1},-{a}_{2}\right)$. When these vectors are added, the sum results in zero.

• $A+\left(-A\right)=\left({a}_{1},{a}_{2}\right)+\left(-{a}_{1},-{a}_{2}\right)=\left({a}_{1}-{a}_{1},{a}_{2}-{a}_{2}\right)=\left(0,0\right)$

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

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.

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

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.

A vector can be moved around along its plane as long as the length/direction does not change.

### 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 $\stackrel{\to }{u}+\stackrel{\to }{w}$:

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 $A=\left(2,3\right)$ and $B=\left(1,3\right)$, 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 $\stackrel{\to }{u}-\stackrel{\to }{w}$, it should be thought of as $\stackrel{\to }{u}+\left(-\stackrel{\to }{w}\right)$. Keeping this in mind, we end up with the figure below:

Fig.7 Parallelogram law for vector subtraction

Let's take some examples.

If$A=\left(2,4\right)$and $B=\left(-2,5\right)$are two vector points, what is the sum of the vectors?

The formula for vector addition is:

$A+B=\left({a}_{1}+{b}_{1},{a}_{2}+{b}_{2}\right)$

The points given are $A=\left(2,4\right)$ and $B=\left(-2,5\right)$

From the points given:

${a}_{1}=2\phantom{\rule{0ex}{0ex}}{a}_{2}=4\phantom{\rule{0ex}{0ex}}{b}_{1}=-2\phantom{\rule{0ex}{0ex}}{b}_{2}=5$

If we substitute in the vector addition formula, we will get:

$A+B=\left(2+\left(-2\right),4+5\right)\phantom{\rule{0ex}{0ex}}=\left(0,9\right)$

If $A=\left(1,7\right)$ and $B=\left(5,-7\right)$ are two vector points, find the sum of the vectors.

The points given are:

$A=\left(1,7\right)\phantom{\rule{0ex}{0ex}}B=\left(5,-7\right)$

$A+B=\left({a}_{1}+{b}_{1},{a}_{2}+{b}_{2}\right)$

From the points we have:

${a}_{1}=1\phantom{\rule{0ex}{0ex}}{a}_{2}=7\phantom{\rule{0ex}{0ex}}{b}_{1}=5\phantom{\rule{0ex}{0ex}}{b}_{2}=-7$

$A+B=\left(1+5,7+\left(-7\right)\right)\phantom{\rule{0ex}{0ex}}=\left(6,0\right)$

Let's take another example.

A toy car moves 10 cm to the east and 24 cm north. Using the triangle law find the resultant vector of the two vectors.

We have two vectors with magnitude 10 cm and 24 cm. Let's call them A and B.

$\stackrel{\to }{A}=10cm\phantom{\rule{0ex}{0ex}}\stackrel{\to }{B}=24cm$

The direction of $\stackrel{\to }{A}$ is the east and the direction of $\stackrel{\to }{B}$ is the north. Therefore, we have:

Fig 8

Notice that the tail of the second vector is placed on the head of the first vector just like the law says. To find the resultant vector, we will complete the triangle by drawing a line to join the tail of the first vector to the head of the second vector and then add both magnitudes.

Let's call the resultant vector C.

Fig. 9

The resultant vector is:

$\stackrel{\to }{C}=10cm+24cm\phantom{\rule{0ex}{0ex}}=34cm$

Let's take another example.

Consider the vectors $\stackrel{\to }{A}=5cm$ in the east direction, $\stackrel{\to }{B}=4cm$ in the north direction and $\stackrel{\to }{C}=7cm$ in the east direction. Using the triangle rule, find the resultant vector.

First, we need to draw the vectors according to their directions. While doing that, keep in mind that the tail of one vector should be placed on the head of another vector.

Fig. 10

As you can see from the figure above, the tail of the second vector is placed on the head of the first vector and the tail of the third vector is placed on the head of the second vector.

The resultant vector will be the summation of the magnitude of all the vectors.

Fig. 11

To find the resultant vector, a line was drawn to connect the tail of the first vector to the head of the third vector. the resultant vector is:

$\stackrel{\to }{C}=5cm+4cm+7cm\phantom{\rule{0ex}{0ex}}=16cm$

Fig. 12

Using the figure above, find $\stackrel{\to }{A}+\stackrel{\to }{B},\stackrel{\to }{B}+\stackrel{\to }{C},\stackrel{\to }{A}-\stackrel{\to }{B}and\stackrel{\to }{B}-\stackrel{\to }{C}$ vectors using the parallelogram law.

Solution

• To find $\stackrel{\to }{A}+\stackrel{\to }{B}$, the parallelogram law can be applied as in the figure. The diagonal of the parallelogram is the sum of the vectors as in the figure below.

Fig. 13

• To find $\stackrel{\to }{A}-\stackrel{\to }{B}$, first vector B should be inversed, and then the parallelogram law should be applied as in the figure below.

Fig. 14

• To find $\stackrel{\to }{B}+\stackrel{\to }{C}$, vector addition can be done with the parallelogram law as in the figure below.

Fig. 15

• To find $\stackrel{\to }{B}-\stackrel{\to }{C}$, first Vector C should be inversed, and then the parallelogram law should be applied as in the figure below.

Fig. 16

## Vector Addition - Key takeaways

• Vector addition can be defined as the procedure of adding two or more vectors.
• Vector addition formula for given points: $A+B=\left({a}_{1}+{b}_{1},{a}_{2}+{b}_{2}\right)$
• 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 sum can be found from the diagonal of that parallelogram.
• Just like regular addition, the order of adding the vectors does not matter.
• Vector subtraction has the same operation as vector addition after inversing the related vectors.

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How do you do graphical vector addition?

Graphical vector addition is done in 1 of 2 ways.

1. Tip-to-Tail method

In this method, you place the tail of one vector to the tip of the other vector. Then, you draw a line starting at the tail of the first vector to the tip of the other vector. This is the resultant vector.

2. Use the parallelogram law

Place the vertices of each vector together. Draw 2 more lines parallel to these vectors, forming a parallelogram. Lastly, draw the diagonal starting at the vertices you placed together. The diagonal is the resultant vector.

Vector addition can be done by using the vector addition formula. The formula is below.

A + B = ( a1 + b1, a2 + b)

Vector addition can also be done graphically and also by using a law called Parallelogram law of vector addition.

What is the vector addition formula?

The vector addition formula is below.

A + B = ( a1 + b1, a2 + b

How do you use the parallelogram law of vector addition?

The parallelogram law of vector addition is used by representing the two vectors to be added as two adjacent sides forming a common vertex and completing it to form a parallelogram. The resultant vector which is the summation can be found from the diagonal of the parallelogram.

Vector addition can be defined as the procedure of adding two or more vectors.

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