Vectors in space

Vectors in Space

Solving a Unit Vector

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The equation for a unit vector is:

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A vector space  is a set that is closed under finite vector addition and scalar multiplication. The basic example is -dimensional Euclidean space , where every element is represented by a list of  real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.

For a general vector space, the scalars are members of a field , in which case  is called a vector space over .

Euclidean -space  is called a real vector space, and  is called a complex vector space.

In order for  to be a vector space, the following conditions must hold for all elements  and any scalars :

1. Commutativity:

(1)

2. Associativity of vector addition:

(2)

3. Additive identity: For all ,

(3)

4. Existence of additive inverse: For any , there exists a  such that

(4)

5. Associativity of scalar multiplication:

(5)

6. Distributivity of scalar sums:

(6)

7. Distributivity of vector sums: