Answer
a)$\vec{\nabla}\cdot\vec{v_a}=0$
b)$\vec{\nabla}\cdot\vec{v_b}=y+2z+3x$
c)$\vec{\nabla}\cdot\vec{v_c}=2x+2y$
Work Step by Step
a)$\vec{\nabla}\cdot\vec{v_a}=(\hat{x}\frac{\partial}{\partial{x}}+\hat{y}\frac{\partial}{\partial{y}}+\hat{z}\frac{\partial}{\partial{z}})\cdot(x^2\hat{x}+3xz^2\hat{y}-2xz\hat{z})=\frac{\partial}{\partial{x}}(x^2)+\frac{\partial}{\partial{y}}(3xz^2)+\frac{\partial}{\partial{z}}(-2xz)=2x-2x=0$
b)$\vec{\nabla}\cdot\vec{v_b}=(\hat{x}\frac{\partial}{\partial{x}}+\hat{y}\frac{\partial}{\partial{y}}+\hat{z}\frac{\partial}{\partial{z}})\cdot(xy\hat{x}+2yz\hat{y}+3xz\hat{z})=\frac{\partial}{\partial{x}}(xy)+\frac{\partial}{\partial{y}}(2yz^2)+\frac{\partial}{\partial{z}}(3xz)=y+2z+3x$
$c)\vec{\nabla}\cdot\vec{v_a}=(\hat{x}\frac{\partial}{\partial{x}}+\hat{y}\frac{\partial}{\partial{y}}+\hat{z}\frac{\partial}{\partial{z}})\cdot(y^2\hat{x}+(2xy+z^2)\hat{y}+2yz\hat{z})=\frac{\partial}{\partial{x}}(y^2)+\frac{\partial}{\partial{y}}(2xy+z^2)+\frac{\partial}{\partial{z}}(2yz)=0+2x+2y=2x+2y$
