Let #\vec A# be the position vector of A and #\vec B# be the position vector of B. A position vector is a vector that points from the origin to a particular point.

If you plot #\vec A#, #\vec B# and #\vec {AB}#, then you can easily notice that #\vec B = \vec A + \vec{AB} # using the triangle rule of addition of vectors. (Try it for two-dimensional vectors!)

In this question, #\vec A = 4\hat i + 2\hat j - 6 \hat k # and #\vec B = 9\hat i - \hat j + 3 \hat k#. So #9\hat i - \hat j + 3 \hat k = (4 \hat i + 2 \hat j - 6 \hat k )+ \vec {AB}#, thus meaning that #\vec {AB} = (9 - 4) \hat i + (-1 - 2) \hat j + (3 + 6)\hat k = 5\hat i - 3\hat j + 9 \hat k#.