The Clear Description on Vector Operations

There are two main types of quantities in physics which are vectors and scalars. Vectors are quantities which have direction and magnitude related to them. And scalars are quantities which only have magnitudes related to them. Scalar quantities can be treated with simple algebra rules. However, this does not apply to vector quantities, they cannot be treated in the same way. Therefore, it is important to know what kind and how many different operations can be executed with these quantities.

Vector

A vector is a quantity that has two independent properties which are magnitude and direction. The term vector also represents the mathematical or geometrical representation of such a quantity.

Vector Quantity

Vector Quantity is a physical quantity which has magnitude with certain unit and also direction. Therefore, specifying the direction of action with its magnitude is essential when we state any vector quantity. Displacement, weight, force, velocity, and more are the examples of vector quantities having direction as well as magnitude both.

Vector Components

A vector mainly has two components, namely the horizontal and the vertical component. The value of horizontal component is cos while the value of vertical component is sin .

Vectors Operations

Since vectors contain directions, these quantities must be treated in such a way that their directions are taken into account. The basic algebra rules do not apply to vectors in general, such as when we simply add the magnitude values ​​of the two vectors, then it gives the wrong result in most cases. 

The following operations are the common operations which performed on vectors in the field of physics:

1. Addition and Subtraction of two Vectors.
2. Multiplication of Vector quantity with Scalar quantity.
3. Product of the two vectors:

1. Dot Product
2. Cross-Product

Addition of Vectors

Vectors cannot be added using the general algebraic rules. When two vectors are added, the magnitude and direction of vectors must be taken into consideration. Vector addition follows the commutative law, which means that the resulting vector is independent from the order in which the two vectors are added.

There are two laws which are used for the addition of vectors. These laws are given as

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

Parallelogram Law of Vector Addition

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

Triangular Law of Vector Addition

When two vectors are arranged in such a manner that the head of one vector is connected with the tail of the second vector then the triangular law of vector addition is used.

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

Product of Vectors

Vectors can be multiplied together, but not divided. There are basically two types of multiplication which are scalar and vector. Dot multiplication (also called scalar product) is a type of multiplication which results in a scalar quantity. Vector multiplication (also termed as cross product) is a type of multiplication which results in a vector quantity. Vector products are used to determine the other derived vector quantities.

Dot Product

Dot product of two vectors is given as

Here,

= angle between two vectors

Cross Product

Cross product of two vectors is given as

The direction of the resulting vector of cross product is determined by the right hand thumb rule.

Vector Subtraction

Vector subtraction is a method of subtracting the coordinates of one vector from the coordinates of another vector. When vectors are subtracted, the direction of vectors which are subtracted must be reversed. This shows that the length of one vector is subtracted from the other vector.

Vector subtraction of 2 (two) vectors a and b is given as a-b. Vector subtraction is the addition of negative of vector b to vector a. This is given as

a-b=a+(-b)

 Thus, vector subtraction contains addition of a vector and the negative of a vector.

Rules for vector subtraction

The rules for vector subtraction are given as:

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

Conclusion

The term vector also represents the mathematical or geometrical representation of such a quantity.

Vector Quantity is a physical quantity which has magnitude with certain unit and also direction. 

Displacement, weight, force, velocity, and more are the examples of vector quantities having direction as well as magnitude both.

The value of horizontal component is cos while the value of vertical component is sin .

Dot product of two vectors is given as

Here,

= angle between two vectors

Cross product of two vectors is given as

There are two laws which are used for the addition of vectors. These laws are given as