Magnitude of a vector

Calculator for the magnitude or length of a four-dimensional vector

Calculate the magnitude of a vector

This function calculates the magnitude of a four-dimensional vector. The magnitude of a vector is the vector's length.

To perform the calculation, enter the vector to be calculated. Then click the 'Calculate' button. Empty fields are counted as 0.


Vector magnitude calculator

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Vector magnitude formula and description

This article describes how to calculate the magnitude of a vector. The magnitude of a vector is its length and can be calculated using Pythagorean theorem. After that, the square of the hypotenuse is equal to the sum of the squares of the legs. The lengths of the legs correspond to the respective coordinates of the vector.

The following figure shows the vector \(\left[\matrix{4\\3}\right]\) in a plane.

The magnitude is the length of the vector, it corresponds to the length of the hypotenuse of a right triangle.

So the length can be calculated:

\(|v|=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\)

The same procedure applies to vectors with more than two dimensions.

\(\left|\left[\matrix{1\\2\\2}\right]\right|=\sqrt{1^2+2^2+2^2}=\sqrt{1+4+4}=\sqrt{9}=3\)
\(\left|\left[\matrix{-4\\6\\-12}\right]\right|=\sqrt{(-4)^2+6^2+(-12)^2}=\sqrt{16+36+144}=\sqrt{196}=14\)
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