Addition of Vectors: Definition, Formula, Laws and Properties

A physical quantity that is represented both in magnitude and direction can be called a vector. The chapter vectors comes under the unit of Vectors and 3-Dimensional Geometry. The entire unit has a weightage. Knowing the vector addition basics helps in solving complex 3D geometry problems. Hence, this article can be considered as a building block for understanding 3 D geometry.

For the additional purposes of these vectors, there are two laws that can be followed. These two laws are:

• Triangle law of vector addition
• Parallelogram law of vector addition

Let us know more about them in detail. 

Keyterms: Vectors, Geometry, Triangle law of vector addition, Parallelogram law of vector addition, Vector Addition

Read More: Determinant Formula

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Triangle law of vector addition

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Let us take two vectors a and b. To add these two vectors, what we’ll do is place the head of any one vector over the tail of the other vector and then join the endpoints of both of these vectors together. This is demonstrated below as we placed the head of the A vector over the tail of the B vector and then joined their endpoints with line c.

Triangle law of vector addition

Line c represents the sum of both the vectors a and b.

Hence, c = a+b

Vector AB is said to be the displacement from A point to B point. The addition of the vectors is commutative in nature, which means if A+B = C, then B+A = C.

Therefore we can write it as

A + B = C = B + A

Hence, the triangle law of vector addition can be written as

\(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}\)

\( \overrightarrow{AB} + - (\overrightarrow{BC})\)

= \(\overrightarrow{a} + \overrightarrow{b}\)

Therefore vector AC represents the difference between vectors a and b. The video below explains this:

Addition of Vectors Detailed explanation:

Also Read:

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Parallelogram Law of Vector Addition

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Let us again take two vectors a and b. To use the parallelogram law of addition, we have to take vector a and b as two adjacent sides of a parallelogram. Then, the diagonal which is formed from the common point of intersection of the two vectors represents the resultant of these two vectors and the sum of them. The direction of these diagonals helps us determine the direction of the final resultant vector. This final resultant vector is shown by C.

This is the parallelogram law of vector addition.

Using the triangle law in the above figure, we can deduce

\(\overrightarrow{OA} = \overrightarrow{AC} + \overrightarrow{OC}\)

Now, since

\(\overrightarrow{AC} = \overrightarrow{OB}\)

\(\overrightarrow{OA} + \overrightarrow{OB} = \overrightarrow{OC}\)

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Properties of Vector Addition

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• Vector Addition is commutative in nature

This means that if we have any two vectors a and b, then for them

\(\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{b} + \overrightarrow{a}\)

• Vector Addition is associative in nature

This means that if we have any three vectors namely a, b and c.

\((\overrightarrow{a} + \overrightarrow{b}) +\overrightarrow{c} = \overrightarrow{a} +(\overrightarrow{b} + \overrightarrow{c})\)

• A zero vector is known as the additive identity for vector addition.

Read More:

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Things to Remember

• A physical quantity that is represented both in magnitude and direction can be called a vector.
• The chapter vectors comes under the unit of Vectors and 3-Dimensional Geometry.
• The entire unit has a weightage.
• There are two laws that can be followed. These two laws are: Triangle law of vector addition, Parallelogram law of vector addition.
• The diagonal which is formed from the common point of intersection of the two vectors represents the resultant of these two vectors and the sum of them.