We define the Euclidean norm of a vector as
By Pythagoras theorem, the Euclidean norm of the vector is equal to the length of the hypothenuse of the right angled triangle with sides and . In other words, the Euclidean norm Euclidean norm of the vector is equal to the distance from the origin to the point , or simply the length of the arrow .
The norm of a vector is .
- if and ;
multiplying a vector by the real number changes the norm of the vector by the factor .
The zero vector has Euclidean norm 0 and if a vector has Euclidean norm 0 then it must be the zero vector.
The Euclidean norm of a vector measures the “length” or “size” of the vector.
There are many possible ways to measure the “size” of a vector corresponding to using different norms. We will meet several alternative norms of a vector below, such as or . We used in the definition of Lipschitz continuity of above.
Example: If then , and .