let's say we start at point A. from point A, we have 2 vectors, one going FROM point A TO point B (direction matters), and one going from point A to point C.

now if we want to know the vector going FROM B to C, we have the following situation:

vector AB = u

vector AC = v

vector AB + vector BC = vector AC (we vector sum by going "head to tail").

that is:

u + vector BC = v

vector BC = v - u.

if we want to find vector CB (start from C, go TO B), we have:

vector AC + vector CB = vector AB

v + vector CB = u

vector CB = u - v.

comparing the two, we see vector BC = -vector CB (same vector, but "opposite sign").

in the picture on the left, we want to find the vector from P to Q. this is what we know:

z + PQ = w (both z + PQ and w start at the origin, and end up at Q).

therefore PQ = w - z.

so why does your book say "z -w"?

because if all we want is the DISTANCE, we take |w - z| = |z - w| (sign doesn't matter for distance).