I know that the following question can be solved with the traditional slope method, but I was thinking of using inner product here, which I learned recently.
Question: If $ x + y = \lambda$ is normal to $y^2 = 12x$ then value of $\lambda$ is?
Here is my approach to the problem. $$ \langle y_1(x) | y_2(x) \rangle = \int y_1(x)y_2(x) dx = 0 $$ $$ y_1(x)y_2(x) = 0 $$ $$ (\lambda - x)\sqrt{x} = 0 $$ Two values are coming from the above equation $x = \lambda, 0$. I don't know what to do from here.
In real case the inner product is defining in an interval such as $[0,1]$ or $[-\pi,\pi]$ and et. You should define inner product exactly equate it zero then you will get an equation that $\lambda $ will be found.