\(\vec{A}\)=Direction Ratio of the Line , \(\vec{B}\)=Position Vector any Point \(B\) on the Line,
\(\vec{C}\)=Position Vector of Point \(C\), \(\vec{C_P}\)=Position Vector of Projected Point \(C_P\) on the Line , \(\vec{C_R}\)=Position Vector of Reflected Point \(C_R\)
\(\vec{CB}=\vec{C}-\vec{B}\), \(\vec{CB}_{||}\)=Projection of Vector \(\vec{CB}\) on Line, \(\vec{CB}_{\perp}\)=Rejection of Vector \(\vec{CB}\) from the Line
\(\vec{C_RB}=\vec{C_R}-\vec{B}\), \(\vec{C_RB}_{||}\)=Projection of Vector \(\vec{C_RB}\) on Line, \(\vec{C_RB}_{\perp}\)=Rejection of Vector \(\vec{C_RB}\) from the Line
Now,
\( \vec{CB}=\vec{CB}_{||} + \vec{CB}_{\perp}\hspace{.6cm}\Rightarrow \vec{CB}_{\perp}= \vec{CB} - \vec{CB}_{||}\) ...(1)
Also,
\( \vec{C_RB}=\vec{C_RB}_{||} + \vec{C_RB}_{\perp}\hspace{.6cm}\Rightarrow \vec{C_RB}_{\perp}= \vec{C_RB} - \vec{C_RB}_{||}\) ...(2)
But Since \(\vec{CB}_{||}=\vec{C_RB}_{||}\), therefore \(\vec{C_RB}_{\perp}= \vec{C_RB} - \vec{CB}_{||}\) ...(3)
Also \(\vec{C_RB}_{\perp}=-\vec{CB}_{\perp}\) (Vectors of Same Length but in Opposite Direction)...(4)
From equations (1), (3) and (4) we have
\(\vec{C_RB} - \vec{CB}_{||}= -(\vec{CB} - \vec{CB}_{||})\)
\(\Rightarrow \vec{C_RB} - \vec{CB}_{||}= -\vec{CB} + \vec{CB}_{||}\)
\(\Rightarrow \vec{C_RB} = 2\vec{CB}_{||} - \vec{CB}\) ...(5)
\(\Rightarrow \vec{C_R}-\vec{B} = 2\vec{CB}_{||} - (\vec{C}- \vec{B})\)
\(\Rightarrow \vec{C_R} = 2\vec{CB}_{||} -\vec{C}+ 2\vec{B}\)
\(\Rightarrow \vec{C_R} = 2(\vec{B} + \vec{CB}_{||}) -\vec{C}\) ...(6)
Also From Fig. (1) we have
\(\vec{C}=\vec{C_P} + \vec{CB}_{\perp}\) (Using Triangle Law of Vectors)\(\hspace{.6cm}\Rightarrow \vec{C_P}= \vec{C} - \vec{CB}_{\perp}\) ...(7)
And \(\vec{C_P}=\vec{B} + \vec{CB}_{||}\) (Using Triangle Law of Vectors)...(8)
From equations (7), (8) we have
\(\vec{B} + \vec{CB}_{||}=\vec{C} - \vec{CB}_{\perp}\) ...(9)
Putting the value of \(\vec{B} + \vec{CB}_{||}\) from equation (9) in equation (6) we have
\(\vec{C_R} = 2(\vec{C} - \vec{CB}_{\perp}) -\vec{C}\)
\(\Rightarrow \vec{C_R} = \vec{C} - 2\vec{CB}_{\perp}\) ...(10)
\(\Rightarrow \vec{C_R} = \vec{C} - 2(\vec{CB}-\vec{CB}_{||})\) ...(11)
\(\Rightarrow \vec{C_R} = \vec{C} - 2 (\vec{CB} - \frac{(\vec{A} \cdot \vec{CB})\vec{A}}{{|\vec{A}|}^2})\) ...(12)
\(\Rightarrow \vec{C_R}=\vec{C} - 2\frac{\vec{A} \times (\vec{CB}\times \vec{A})}{{|\vec{A}|}^2}=\vec{C} - 2\frac{(\vec{A} \times \vec{CB})\times \vec{A}}{{|\vec{A}|}^2}\) ...(13)
Please Note that equations (6), (10), (11) and (12) give the formula for Reflection in 2D, 3D and Higher Dimensions.
Since Cross Product is only applicable in 2D and 3D Vectors, equation (13) gives the formula for Reflection in 2D and 3D only.