# Norm of a Vector

We define the Euclidean norm $\vert a\vert$ of a vector $a=(a_1,a_2)\in R^2$ as

• $\vert a\vert =(a_1^2+a_2^2)^{1/2}$.

By Pythagoras theorem, the Euclidean norm $\vert a\vert$ of the vector $a=(a_1,a_2)$ is equal to the length of the hypothenuse of the right angled triangle with sides $a_1$ and $a_2$. In other words, the Euclidean norm Euclidean norm of the vector $a=(a_1,a_2)$ is equal to the distance from the origin to the point $a=(a_1,a_2)$, or simply the length of the arrow $(a_1,a_2)$.

The norm $\vert a\vert$ of a vector $a=(a_1,a_2)$ is $\vert a\vert=(a_1^2+a_2^2)^{1/2}$.

We have

• $\vert \lambda a\vert =\vert\lambda\vert \vert a\vert$ if $\lambda\in R$ and $a\in R^2$;

multiplying a vector by the real number $\lambda$ changes the norm of the vector by the factor $\vert\lambda\vert$.

The zero vector $(0,0)$ 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 $a=(a_1,a_2)$ below, such as $\vert a_1\vert +\vert a_2\vert$ or $\max(\vert a_1\vert, \vert a_2\vert )$. We used $\vert a_1\vert +\vert a_2\vert$in the definition of Lipschitz continuity of $f: R^2\rightarrow R$ above.

Example: If $a=(3,4)$ then $\vert a\vert =\sqrt{9+16}=5$, and $\vert 2a\vert =10$.