Vector Addition and Subtraction

Introduction

Vector is defined as a quantity that has both magnitude and direction. Examples for vector quantity are displacement, velocity, acceleration, force, etc. Vector subtraction and addition does not follow the simple arithmetic rules. A special set of rules are used for vector subtraction and addition. 

Vector Addition 

The process of addition of two or more vectors together is known as vector addition. Depending on the direction of the vector, vector addition is classified into two types:

1. Parallelogram Law of Vector Addition
2. Triangular Law of Vector Addition

Vector addition method is chosen depending on the arrangement of the head and tail of vectors.

• When two vectors are arranged head to tail then the triangular law of vector addition is applicable
• When two vectors are arranged head to head or tail to tail then the vector addition is performed using parallelogram law

Parallelogram Law of Vector Addition

When two vectors are arranged head to head or tail to tail then the vector addition is performed using parallelogram law.

According to the Parallelogram Law of Vector Addition when two vectors are supposed to be the adjacent sides of parallelogram then the resultant of the two vectors is the vector which passes diagonally through the point of contact of the two vectors.

Triangular Law of Vector Addition

When two vectors are arranged head to tail then the triangular law of vector addition is applicable.

According to the Triangular Law of Vector Addition If two vectors are represented by two sides of a triangle taken in order of magnitude and direction, then the third side of that triangle defines the magnitude and direction of the resultant vector.

Why Addition of vectors important?

In physics, vector quantities such as force interact with each other and create a resultant effect on the objects to which they are applied. Because the impact/influence of all these forces is taken into account when determining the nature of the system’s motion, operations such as addition, subtraction, and multiplication of these forces are required to find the resultant of these forces.

Vector Addition Examples

If p = {4, -2, 2} and q = {1, -3, 2}. Determine p+q

Ans.

p+q = {4,-2,2} – {1,-3,2}

p+q = {(4+1), -2+(-3), 2+2}

p+q = {5,-5,4}

Vector Subtraction

Vector subtraction is defined as a process of subtracting the coordinates of one vector from the coordinates of another vector. When subtracting vectors, the direction of the vector to be subtracted must be reversed. This indicates that the length of one vector is subtracted from the other vector.

Vector subtraction of 2 (two) vectors a and b is defined by a – b . It is the addition of negative of vector b to vector a that is a – b = a + (-b) . Hence, vector subtraction includes the addition of vectors and the negative of a vector.

Rules for vector subtraction

The rules for vector subtraction are:

1. Vector subtraction is done only between two vectors.
2. Both vectors in the vector subtraction will have the same physical quantity.
3. For vector subtract of two vectors a and b graphically (to determine a – b) we just need to make them coinitial first and then we have to draw a vector from tip of b to tip of a.
4. We can add -b (negative of vector b which obtained through multiplying b with -1) to a so as to execute the vector subtraction  that is a – b = a + (-b) .
5. When the vectors are in component form then we just need to subtract their respective components in the order of vector subtraction.

Properties of Vector Subtraction

• When a vector is subtracted from itself then it is known as zero vector, that is 

        p -q =0  (for vector p)

• The vector subtraction is not commutative, that is p – q may or may not equal to q – p 
• Vector subtraction is also not associative, that is (p – q) – r may or may not equal to p – (q – r) 

Conclusion

Vector is defined as a quantity that has both magnitude and direction. Examples for vector quantity are displacement, velocity, acceleration, force, etc. 

Vector subtraction and addition does not follow the simple arithmetic rules. A special set of rules are used for vector subtraction and addition. 

According to the Triangular Law of Vector Addition If two vectors are represented by two sides of a triangle taken in order of  magnitude and direction , then the third side of that triangle defines the magnitude and direction of the resultant vector.

When a vector is subtracted from itself then it is known as zero vector. That is p – p = 0  (for vector p).

The vector subtraction is not commutative. That is p – q may or may not equal to q – p 

Vector subtraction is also not associative. That is (p – q) – r may or may not equal to p – (q – r) .