Vectors: Components and Magnitude

A vector from initial point $(a,b)$ to terminal point $(c,d)$ has components $\langle c-a,\; d-b \rangle$ — terminal minus initial, in that order.

Its magnitude is $|v| = \sqrt{(c-a)^2 + (d-b)^2}$. Contest questions almost always ask for $|v|^2$ so the answer stays an integer.

For a sum, add componentwise first and take the magnitude once; never take two square roots where one will do.