One of the differences between vectors and points is that while it doesn't really make sense to add or subtract points, it does make sense to add and subtract vectors. Here's how:
To add two vectors , we consider as the displacement vector for some P and then as the displacement vector for some Q. The answer, is then the displacement vector .
The following picture proves that vector addition is commutative; i.e., that :
To define how to subtract two vectors, it suffices to define how to negate a single vector, as . This is easy enough: is simply the vector with the same magnitude as , but with the opposite direction. To put it another way, if for some point P, then .
Finally, we can define how to multiply a vector by a scalar real number: for some non-negative real number c is the vector with the same direction as , but with c times 's magnitude. This definition of scalar multiplication has all of the properties you would expect it to have: and more generally .
Fortunately, all of these operations are easy to calculate in Cartesian coordinates. The trick is to realize that (v1, v2, v3) and are the same vector. Why? Because they have the same projections onto the coordinate axes. Once we have this knowledge, we can see that , or that coordinate-wise, vector addition is the same as tuple addition.
It should then come as no surprise that multiplying a vector by a scalar is the same as multiplying a tuple by a scalar: . To prove this, let for some point P and let for some point Q. By hypothesis, O, P, and Q all lie on the same line. If we project P and Q onto the XY plane and call the results P' and Q respectively, then we get two similar triangles: . We are given that , so by the Similar Triangles Theorem . But , so the Z component of c ( v1, v2, v3 ) must be c v3. The same reasoning holds for the X and Y components, so the result follows.