Question 11
Let i,j and k be three unit vectors along three mutually perpendicular axes, namely, the x,y and z axes respectively, and the origin O is the intersection of the three axes. A plane mirror in the space has equation $\mathbf r\cdot(\mathbf i+2\mathbf j+3\mathbf k)=5$. A ray of light is shone along the path with equation $\mathbf r=5\mathbf i+2\mathbf j+\mathbf k+\lambda(5\mathbf i+\mathbf j)$, where λ is real. The ray of light hits the plane and is reflected along the path $\mathbf r=\mathbf j+\mathbf k+\mu(4\mathbf i+b\mathbf j+c\mathbf k)$, where μ is real. Assuming that the incident ray, reflected ray and the normal at the point of incidence lie on the same plane, and that the incident ray and the reflected ray make the same angle with the mirror, find the value of |b|+|c|.