Finding a vector given its dot products with given vectors Given vectors $\vec{A} = -4.7\hat{\imath}  + 8.2\hat{\jmath}$  and $\vec{B}= 9.6\hat{\imath} + 6.7\hat{\jmath}$, determine the vector C that lies in the $xy$-plane perpendicular to $\vec{B}$ and whose dot product with $\vec{A}$ is $20.0$.

Can someone walk me through the steps of solving this problem? I need to solve for $\vec{C} = ?\hat{\imath} + ?\hat{\jmath}$
Instead of "?", you can fill in the missing components with letters, say $x$ and $y$.  Suppose that $\vec C=x\hat i+y\hat j$.  Then $\vec C\cdot \vec A=-4.7 x+8.2y$.  What must the last expression equal? Similarly, write down an expression for $\vec C\cdot \vec B$ in terms of $x$ and $y$ to get an equation based on the assumptions in the problem.  Then try to solve the system of equations.
Hint:
Vectors perpendicular to $\vec B = 9.6 i + 6.7j$ are of form $k (6.7 i - 9.6 j) $ for some scalar $k$.
Can you work out why and find $k$ from the condition on dot product with $\vec A$?