Solution verification for problem in vectors.

Question

Let $V_1=ai+bj,V_2=ci-dj$, vector in $\Bbb R^2$, where $a= x^3y/2,b=x+2y,c=2x-3y,d=y$.

$V_1 \cdot V_2 = ac-bd$.

$\nabla \cdot V_1 = 3x^2y/2+2$.

Component of $V_1$ normal to $V_2$: $V_1 - \text{proj}_{V_2}(V_1 ) = ai+bj - \frac{ac-bd}{c^2+d^2}(ci-dj)$

gradient of a: $(3x^2y/2,x^3/2)$

Is the solution correct?

Answer 1

Yes, your solutions seems correct to me.