Suppose M is the midpoint of Segment AB, P is the midpoint of Segment AM, and Q is the midpoint of segment PM.

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The coordinates of M, P and Q in terms of a and b are $M = \frac{1}{2}\cdot a + \frac{1}{2}\cdot b$ , $P = \frac{3}{4}\cdot a + \frac{1}{4}\cdot b$ and $Q = \frac{1}{8}\cdot a - \frac{1}{8}\cdot b$ , respectively.

In this question we are going to use definitions of vectors and product of a vector by a scalar. Based on the information given on statement, we have the following vectorial formulas:

Location of M

$\overrightarrow{AM} = \frac{1}{2}\cdot \overrightarrow{AB}$

$\vec M - \vec A = \frac{1}{2}\cdot \vec B - \frac{1}{2}\cdot \vec A$

$\vec M = \frac{1}{2}\cdot \vec A +\frac{1}{2}\cdot \vec B$

$M = \frac{1}{2}\cdot a + \frac{1}{2}\cdot b$

Location of P

$\overrightarrow{AP} = \frac{1}{2}\cdot \overrightarrow{AM}$

$\vec P - \vec A = \frac{1}{2}\cdot \vec M - \frac{1}{2}\cdot \vec A$

$\vec P = \frac{1}{2}\cdot \vec A +\frac{1}{2}\cdot \vec M$

$\vec P = \frac{3}{4}\cdot \vec A + \frac{1}{4}\cdot \vec B$

$P = \frac{3}{4}\cdot a + \frac{1}{4}\cdot b$

Location of Q

$\overrightarrow{QM} = \frac{1}{2}\cdot \overrightarrow{PM}$

$\vec M - \vec Q = \frac{1}{2}\cdot \vec M - \frac{1}{2}\cdot \vec P$

$\vec Q = \frac{1}{2}\cdot \vec P - \frac{1}{2}\cdot \vec M$

$\vec Q = \frac{1}{2}\cdot \left(\frac{3}{4}\cdot \vec A + \frac{1}{4}\cdot \vec B\right) -\frac{1}{2}\cdot \left(\frac{1}{2}\cdot \vec A + \frac{1}{2}\cdot \vec B\right)$