Length of a vector

Online calculator for calculating the length of a vector with 3 elements

Calculate the length of a vector

This function calculates the length of a vector To perform the calculation, enter the vector to be calculated and click the Calculate button. Empty fields are counted as 0.


Vector length calculator

VectorResult
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Decimal places

Description and formula to calculate the length of a vector

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 elements.

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

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