Find the angle between the planes x + y + z = 1 and x - 2y + 3z = 1.

Concept: 

The angle between the two planes A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0 is

i.e. \(\theta = \cos^{-1} \left( \frac{A_1A_2+B_1B_2+C_1C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}} \right) \)

Calculation: 

Given: The planes x + y + z = 1 and x - 2y + 3z = 1.

Here  A1 = 1,  B1 = 1 C1 = 1 and A2 = 1, B2 = -2 C2 = 3

∴ The angle between the two planes x + y + z = 1 and x - 2y + 3z = 1.is

\(\theta = \cos^{-1} \left( \frac{A_1A_2+B_1B_2+C_1C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}} \right) \)

\(\theta = \cos^{-1} \left( \frac{1(1)+1(-2)+1(3)}{\sqrt{1^2+1^2+1^2}\sqrt{1^2+(-2)^2+3^2}} \right) \)

\(\theta = \cos^{-1} \left( \frac{1-2+3}{\sqrt{3}\sqrt{14}} \right)\) ⇒ \(\theta = \cos^{-1} \left( \frac{2}{\sqrt{42}} \right)\) ⇒ \(\theta = \rm \cos^{-1}\left(\sqrt{\frac{2}{21}}\right) \)

∴ The angle between the planes is \( \rm \cos^{-1}\left(\sqrt{\frac{2}{21}}\right) \) .