Q:

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 $\vec{C} = ? \hat{\imath} + ? \hat{\jmath}$

A:

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.