Problem
Let $x$ and $y$ be real numbers and consider the column vectors $u=\left[\begin{array}{c}2 \\ x \\ -3\end{array}\right]$ and $v=\left[\begin{array}{c}3 \\ -4 \\ -y\end{array}\right]$ If the inner product of $u$ and $v$ is -5 , then what is $3 y-4 x$ ?
Solution
Step 1 ：Find the inner product of u and v: \(2*3 + x*(-4) + (-3)*(-y) = -5\)
Step 2 ：Solve for x in terms of y: \(x = \frac{3y}{4} + \frac{11}{4}\)
Step 3 ：Substitute x into the expression \(3y - 4x\) and simplify: \(3y - 4\left(\frac{3y}{4} + \frac{11}{4}\right) = -11\)
Step 4 ：\boxed{-11}
