Let's say we have the folowing hyperplane $$\{x|a^{T}x= b\}$$ where $a\in \Re^n, a\neq 0, b\in \Re$ as the set of points with a constant inner product to a given vector a. I want to prove that if the line: $$l(t)=x+t\cdot a$$ where $t\in \Re$ intersects the hyperplane at $x_1$, then $x_1=\big(\frac{b}{||a||^2}\big)a$ where Euclidean norm is implied.
I am confused. Any clue?
Pluggin the line into the hyperplane gives you $b=a^T\cdot(x+ta)=a^T\cdot x+ta^T\cdot a=a^T\cdot x+t\Vert a\Vert^2$ and thus $t=\frac{b-a^T\cdot x}{\Vert a\Vert^2}$. Plugging this into the line gives $x_1=x+\frac{b-a^T\cdot x}{\Vert a\Vert^2}a=x+\frac{b}{\Vert a\Vert^2}a-\frac{a^T\cdot x}{\Vert a\Vert^2}a=\frac{b}{\Vert a\Vert^2}a$.
Edit: As Aretino pointed out my computation is incorrect. See my comment below which gives you a counter-example to the original question's assumption.