\(\vec{A}\)=Direction Ratio of the Normal , \(\vec{B}\)=Position Vector any Point \(B\) on the Line/Plane/Hyper-Plane,
\(\vec{C}\)=Position Vector of Point \(C\) , \(\vec{C_R}\)=Position Vector of Reflected Point \(C_R\)
\(\vec{CB}=\vec{C}-\vec{B}\), \(\vec{CB}_{||}\)=Projection of Vector \(\vec{CB}\) on Normal, \(\vec{CB}_{\perp}\)=Rejection of Vector \(\vec{CB}\) from the Normal
\(\vec{C_RB}=\vec{C_R}-\vec{B}\), \(\vec{C_RB}_{||}\)=Projection of Vector \(\vec{C_RB}\) on Normal, \(\vec{C_RB}_{\perp}\)=Rejection of Vector \(\vec{C_RB}\) from the Normal
Now,
\( \vec{CB}=\vec{CB}_{||} + \vec{CB}_{\perp}\hspace{.6cm}\Rightarrow \vec{CB}_{||}= \vec{CB} - \vec{CB}_{\perp}\) ...(1)
Also,
\( \vec{C_RB}=\vec{C_RB}_{||} + \vec{C_RB}_{\perp}\hspace{.6cm}\Rightarrow \vec{C_RB}_{||}= \vec{C_RB} - \vec{C_RB}_{\perp}\) ...(2)
But Since \(\vec{CB}_{\perp}=\vec{C_RB}_{\perp}\), therefore \(\vec{C_RB}_{||}= \vec{C_RB} - \vec{CB}_{\perp}\) ...(3)
Also \(\vec{C_RB}_{||}=-\vec{CB}_{||}\) (Vectors of Same Length but in Opposite Direction)...(4)
From equations (1), (3) and (4) we have
\(\vec{C_RB} - \vec{CB}_{\perp}= -(\vec{CB} - \vec{CB}_{\perp})\)
\(\Rightarrow \vec{C_RB} - \vec{CB}_{\perp}= -\vec{CB} + \vec{CB}_{\perp}\)
\(\Rightarrow \vec{C_RB} = 2\vec{CB}_{\perp} - \vec{CB}\)
\(\Rightarrow \vec{C_RB}= 2(\vec{CB}-\vec{CB}_{||}) - \vec{CB}\)
\(\Rightarrow \vec{C_RB}= \vec{CB}-2\vec{CB}_{||}\) ...(5)
\(\Rightarrow \vec{C_R}-\vec{B} = \vec{C}-\vec{B}-2\vec{CB}_{||}\)
\(\Rightarrow \vec{C_R} = \vec{C}-2\vec{CB}_{||}\) ...(6)
\(\Rightarrow \vec{C_R} = \vec{C} - 2(\frac{(\vec{A} \cdot \vec{CB})\vec{A}}{{|\vec{A}|}^2})\) ...(7)
Please Note that equations (6) and (7) give the formula for Reflection in 2D, 3D and Higher Dimensions.