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\vertin 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.

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