When using vector notation in coordinate systems (Cartesian coordinates) we see that the magnitude of a vector in two dimensions is equal to the square root of its Y component squared added to its X component squared (Pythagorean theorem).
But the same calculation is done for a three dimensional vector that has X, Y, and Z components.
Is there a triangle that has four sides? (of course not, but how does this right triangle formula work for a calculation that involves more than two dimensions?).
[Math] Pythagorean theorem in higher dimensions
geometryvectors
Best Answer
You can think of it as doing the pythagorean theorem twice. Imagine you have the vector (x,y,z) denoted by the red line in the figure below. The magnitude of the green line is given by
$\sqrt{x^2+y^2}$
and the magnitude of the blue line is $z$. So when you use the pythagorean theorem on the triangle made up of the red, green, and blue lines you get
$\sqrt{\sqrt{x^2+y^2}^2 + z^2} = \sqrt{x^2+y^2+z^2}$