Vectors

A vector is an element of a vector space, which is a collection of vectors.

In the diagram below, (x,y) and (x,y,z) are vectors.

Vector Magnitude

The magnitude of a vector is equal to its length. This is calculated as the square root of the sum of the squares of its individual coordinates:

$\\\left \| a \right \| = \sqrt{a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}} \\\\\textup{where:} \\\\\left \| a \right \|\,\,\textup{magnitude of vector a} \\a_{n}\,\,\,\,\,\,\textup{individual coordinates of vector a} \\n\,\,\,\,\,\,\,\,\,\textup{number of coordinates of vector a}$

Vector Dot Product

A vector dot product:

is an algebraic operation that takes two equal-length sequences of numbers
returns a single number
derives its name from the dot symbol used to designate the operation
is the sum of the products of the corresponding entries of the two sequences of numbers

The equation for calculating a dot product is shown below:

$\\a\cdot b = \sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}\,+\,\cdots\,+a_{n}b_{n}=c \\\\\textup{where:} \\\\a\,\,\,\,\,\,\textup{vector of values} \\b\,\,\,\,\,\,\,\textup{vector of values} \\c\,\,\,\,\,\,\,\textup{single value result} \\n\,\,\,\,\,\,\textup{number of values in a and b} \\i\,\,\,\,\,\,\,\textup{iterator over vector values}$

An example is:

$\\ \textup{given:} \\a=\left [ -2,1,4 \right ] \\b=\left [ 2,1,2 \right ] \\\\\textup{then:} \\a\cdot b = \sum_{i=1}^{3}a_{i}b_{i}=\left ( -2*2 \right )+\left ( 1*1 \right )+\left ( 4*2 \right )=5$

Euclidean Vector

A Euclidean Vector is a geometric object that has a magnitude and direction.

Vectors

Vector Magnitude

Vector Dot Product

Euclidean Vector

References