Chapter 13 - Vector Geometry - 13.1 Vectors in the Plane - Exercises - Page 650: 52

$a=0$ and $b=4$.

Since the figure is a parallelogram, the following two vectors must be equivalent: $\left( { - 1,b} \right) - \left( { - 3,2} \right) = \left( {2,b - 2} \right)$ ${\ \ \ }$ and ${\ \ \ }$ $\left( {2,3} \right) - \left( {a,1} \right) = \left( {2 - a,2} \right)$. Thus, $\left( {2,b - 2} \right) = \left( {2 - a,2} \right)$. So, $a=0$ and $b=4$.