[justify]For the task of finding the Pythagorean formula for the magnitude of the vector, two triangles are seen, the first one helps deducing that [math]A_{xy}^2+A_z^2=|\vec{A}|^2[/math], and the second one is the triangle formed on the xy plane which allows the deduction of the relation [math]A_{xy}^2=A_x^2+A_y^2[/math],  thus combining the latter equations to get  [math]A_x^2+A_y^2+A_z^2=|\vec{A}|^2[/math] is straightforward. Both triangles are shown in the following figures:[/justify]