$$\therefore \bar{A} = \frac{\partial}{\partial x}(xy + yz + zx)\bar{i} + \frac{\partial}{\partial y}(xy + yz + zx)\bar{j} + \frac{\partial}{\partial z}(xy + yz + zx)\bar{k}$$ $$\therefore \bar{A} = (y + z)\bar{i} + (x + z)\bar{j} + (y + x)\bar{k}$$ $$\therefore a_{1} = (y + z), a_{2} = (x + z), a_{3} = (y + x)$$ $$\therefore \bar{V}\cdot \bar{A} = \frac{\partial}{\partial x}a_{1} + \frac{\partial}{\partial y}a_{2} + \frac{\partial}{\partial z}a_{3} = 0 + 0 + 0 = 0$$ $$\bar{V} \times \bar{A} = \begin{vmatrix} i & j & k \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y + z & x + z & y + x \end{vmatrix} \ \therefore\bar{V} \times \bar{A} = i\bigg[\frac{\partial}{\partial y}(x + y) - \frac{\partial}{\partial z}(x + z)\bigg] - j\bigg[\frac{\partial}{\partial x}(y + x) - \frac{\partial}{\partial z}(x + z)\bigg] + k\bigg[\frac{\partial}{\partial x}(x + z) - \frac{\partial}{\partial y}(y + z) \bigg]$$ $$\therefore \bar{V} \times \bar{A} = i[1 - 1] - j[1 - 1] + k[1 - 1]$$ $$\therefore \bar{V} \times \bar{A} = 0$$