Finding the magnitude of a vector

I had a question earlier about finding the magnitude of a vector. The easiest general way to find the magnitude of a vector is to take the scalar product of the vector and itself, since the angle between a vector and itself is zero:

\vec{A}\cdot\vec{A} = |\vec{A}||\vec{A}|\cos{\theta} = |\vec{A}|^2

If you know the vector in x-y component form,

\vec{A}\cdot\vec{A} = \left(A_x\,\hat{\imath}+A_y\,\hat{\jmath}\right)\cdot\left(A_x\,\hat{\imath}+A_y\,\hat{\jmath}\right) = A_x^2 + A_y^2 = |\vec{A}|^2

It works out this way because the unit vectors are orthogonal (perpendicular) in our cartesian system:

\hat{\imath}\cdot\hat{\imath}=1, \quad \hat{\imath}\cdot\hat{\jmath}=0 

If you do it this way, it will work in all coordinate systems. Treating the vector as a little right triangle only works if you know the vector in x-y component form.