Find the angle between the planes x + 2y + z = 7 and 2x – y + z = 13.

CONCEPT:

• Let A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂ z + D₂ = 0 are the equations of two planes aligned at an angle θ where A₁, B₁, C₁ and A₂, B₂, C₂ are the direction ratios of the normal to the planes, then the cosine of the angle between the two planes is given by:

\(cos\theta = \left| {\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}} \right|\)

CALCULATION:

Given planes are x + 2y + z = 7 and 2x – y + z = 13.

\(cos\theta = \left| {\frac{{1.2 - 2.1 + 1.1}}{{\sqrt {{1^2} + {2^2} + {1^2}} \sqrt {{2^2} + {{\left( { - 1} \right)}^2} + {1^2}} }}} \right| = \frac{1}{6}\)