How this calculator works
The Euclidean magnitude extends the Pythagorean theorem to every component. Signs disappear when components are squared, so reversing a vector does not change its length. The tool accepts two, three, four or five coordinates and treats them as ordinary Euclidean components. A missing component must supply the part of the squared length not already supplied by the known coordinates.
Formula
|v| = sqrt(v_1^2 + ... + v_n^2). For missing component x, x = plus or minus sqrt(|v|^2 - sum(known_i^2)).
Worked example
(3, 4, 12) has magnitude sqrt(9 + 16 + 144) = 13. For a planar vector of magnitude 5 whose first component is 3, the other component is either 4 or -4.
Advertisement
Frequently asked questions
Can magnitude be negative?
No. A magnitude is a distance and is always nonnegative. The zero vector has magnitude zero.
Is a fourth component treated as time?
No. All entries use the Euclidean sum of squares. A relativistic spacetime interval uses a different metric and is outside this tool.
What units should the components use?
Use the same unit for every component; the magnitude has that unit. Dimensionless coordinates give a dimensionless magnitude.