A vector is defined as a special mathematical structure that is used in mathematics and in geometric applications; it is specifically defined as a quantity with the magnitude and **direction** .Vectors have the ability to find the directions of defined **object** with their magnitude. As an example wind is defined in terms of both speed as well as **direction** or if take an example of moving **object** then it also defined or express in forces applies on that body and the **direction** of the applied force .On a Cartesian coordinate plane positions of the **object** or locations of points are expressed in terms of ordered pair ( x , y ) that is a particular example of vector .For a vector ( x , y ) is defined as a certain distance that is a magnitude from and an angle that is defined the **direction** related to the origin ( 0 , 0 ).For simplifying the problems related to the three – dimensional geometry vectors are very useful .

In simple terms scalar are **called** as real numbers and vectors are **called** as of dimension n that is an ordered collection of elements n , that are **called** as components . Scalar quantities often represent as the variables just as we define them in algebra but when we talk about the vector quantities than they are represented in form of bold and capital letters like **X.** A vector can also be represented with a right – handed arrow. If an n – dimensional vector **X** has n elements then it is denoted symbolically as:

**X = X1 , X2 , . . . . , Xn > or X = ( X1 , X2 , . . . . , Xn )**

The above representation is understood in terms of numerical values as ( 2 , -5 ) , ( -1 , 0 , 2 ) and

( PI , a , b , 2 / 3 ) , these are the examples of vectors that have the dimensions 2 , 3 , 4 respectively .

If two corresponding components are **equal** then their vectors are also **called** as the **equal** vectors.

When we talking about the Vector Calculator it is very useful online tool for the students .by this tool students get the answers of their values without delay .In vector there are several operations like addition , subtraction , multiplication etc that are also supported by the Vector calculator like Vector addition calculator or cross product calculator etc. In the vector calculatoruser can enter the real number as well as the complex number for carrying the calculations. Let’s take an example of calculation by the vector calculator for a subspace E of **X**4 that is generated by two of the vectors

V1 = ( 1 , 2 , 3 , 4 ) and V2 = ( 1 , -1 , -1 , 1 ) and V1 , V2 are the orthogonal components correspond to the linear system then define the E . There are another two vectors v1 = ( -5 , -2 , 3 , 0 ) and v2 = ( -2 , -5 , 0 , 3 ) .

Then by the linear system, E is defined as:

-5 x1 – 2 x2 + 3 x3 = 0 ,

-2 x1 – 5 x2 + 3 x3 = 0

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