The length of a vector is also known as its magnitude, and is computed using Pythagoras' Theorem.

For 2D Vectors this is given as follows:

• $|~(x,y)~|~=~\sqrt{x^2+y^2}$
For 3D Vectors this is given as follows:

• $|~(x,y,z)~|~=~\sqrt{x^2+y^2+z^2}$
For higher dimensional vectors we have the generalised formula:

• $|~(x_1,x_2,\ldots,x_n)~|~=~\sqrt{x_1^2+x_2^2+\ldots+x_n^2}$
Just as the absolute value of a real number (or rational number, etc) can be computed by taking the (positive) square root of the square of the number, the magnitude of a vector is the square root of the scalar product with itself:

• $|v|~=~\sqrt{v.v}$

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