So vectors and operations on them. Scalar multiplication acts on single vectors and changes the magnitude. Take pairs of vectors, in physics of course, these have to have the same units if you want to add them together or "take" the inner product. Referring back to a previous diagram Please Register or Log in to view the hidden image! If the vector P is also a position vector (rotating or fixed), then it's the sum of various other pairs of coordinate vectors present in the diagram. For instance \( P = p^1 + p^2 \). Another way to write this is \( P = p^i \) with implied summation. If the projection \( P_1 \) were added to the dashed line from the end of P onto X1 (maybe think of this line as a projector), the sum would also equal P. By adding the coordinate line \( \Xi^2 \) perpendicular to X1 and parallel to the projected line, you then have a real component to add. Note also that being parallel to implies being translated from, such that a direction is left the same . . .