Vector Addition using Components

With vectors written as $\vec A = A_x\hat i+A_y\hat j+A_z\hat k$ and $\vec B = B_x\hat i+B_y\hat j+B_z\hat k$, adding (or subtracting) the two vectors is the same as adding (or subtracting) their matching components:

This component method is completely equivalent to the geometric triangle/parallelogram law, but is usually far quicker for numerical problems — no angles or trigonometric identities are needed, only ordinary addition/subtraction of numbers, axis by axis. One subtlety: the numerical values of the components depend on the coordinate system chosen (rotate the axes and the components change), but the geometric result of adding or subtra …