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A vector is defined as an object which has both a direction and a magnitude. It is considered that two vectors are the same if both have the same direction and magnitude. 

Operations on Vectors

Some basic operations can be performed on vectors geometrically without any coordinate system as a reference. Vector operations are given by a scalar in the form of addition, subtraction, and multiplication. There are two different ways of multiplication of two vectors together, the dot product and the cross product. The operations on vectors are mentioned below:

Addition of vectors: To get the final value the individual components of the respective vectors are added.

Subtraction of vectors: This is similar to the addition of vectors. Here, the sign of one of the vectors is changed in direction and added to the other vector.

Scalar Multiplication of vectors: A scalar is a real number that does not have any direction. So at the time of multiplication, the scalar is multiplied with each component of the vector. The operation of multiplying a vector by a scalar is known as scalar multiplication.

Scalar Triple Product of vectors: The scalar Triple Product of vectors means the dot of one vector with the cross product of the other two vectors. In this operation, if any two vectors are equal, then the result will be zero.

Multiplication of vectors: There are two ways of multiplication of vectors.

• Dot Product of Vectors, that means the individual vectors are multiplied, and then the result is added to get the dot product.
• Cross Product of Vectors means the components of the vector are shown in a matrix and it represents the final result of the cross product of vectors.

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Types of Vectors

Here few types of vectors are explained briefly with their properties:

1. Zero Vectors: Vectors with zero magnitudes are called zero vectors. It has no direction and zeroes magnitudes. It is also known as the additive identity of vectors. 
2. Unit vectors: Vectors having a magnitude equal to one are known as unit vectors. The magnitude of these kinds of vectors is one, and these are used for denoting direction. It is also known as the multiplicative identity of vectors.
3. Position vectors: In a three-dimensional space position and direction of movement of vectors are determined by position vectors. It is also termed a location vector.
4. Equal Vectors: Direction and magnitude both are equal for equal vectors. If corresponding components are equal then two or more vectors are said to be equal.
5. Negative vector: If one of the two vectors has the same magnitudes but directions are opposite then only a vector is said to be negative of another vector.
6. Parallel Vectors: Two vectors having the same directions but not necessarily the same magnitude are termed parallel vectors.
7. Orthogonal vectors: If two vectors are placed in a space and the angle between them is 90 degrees, then they are called orthogonal vectors.
8. Co-initial vectors: Co-initial vectors are those that have some initial points.

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