we need to find unit vector along the reflected ray.

Question

A ray of light on a plane mirror comes along a vector $i+j-k$

The normal on incidence point is along $i+j$

we need to find unit vector along the reflected ray.

I am not able to solve and draw the picture. could any one explain me what is going on?

answer given is $-{1\over\sqrt{3}}(i+j+k)$

Thank you for helping.

Answer 1

Let's say that $u$ is the vector of incoming ray, $n$ is the normal, and $v$ is the reflected ray. The normal component of $u$ is $$\frac{u\cdot n}{|n|^2}n$$ This is the component that must be reversed. We reverse it by subtracting twice of it: $$v = u - 2\,\frac{u\cdot n}{|n|^2 } n$$ is the reflected vector. Here $$v = i+j-k -2\,\frac{2}{2}(i+j) = -i-j-k$$ which you are then asked to normalize.