A vector is a container where order matters and repititions are allowed. An N-vector has n components (elements), each component called Different values in a vector can be the same, so:
is entirely possible with Vectors.
A 2-vector represents 2 dimensional space and a 3-vector represents 3-dimensional space.
Two vectors can be added like so:
The resultant is a new vector, called W.
You can only add two vectors if the vectors have the same number of components.
Scalar multiplcation is where you increase every item in a vector by R. Let R be a real number then:
Vector length can be defined using cardinality. The length of a vector, A, is |A|. The length of a vector has uses this formula:
If you try to calculate the length of a negative vector, you get the length of the positive vector.
Another thing to note is:
To reverse a vector you times each component by -1, making everything negative, and
Start with some set of numbers such as real numbers, integers, whole numbers, rationals etc.
The superscript number ontop of a set of numbers such as is the set of all real valued (numbers that are in the real numbers) n-vectors where each component is in the set of Real numbers.