The angle between the planes 2x + y + z = 7 and x - y + 2z = 9 is:

Concept:

The acute angle θ between two planes a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂ is given by the formula:

\(\rm \cos \theta = \dfrac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{(a_{1}^2+b_{1}^2+c_{1}^2)(a_{2}^2+b_{2}^2+c_{2}^2)}}\).

Calculation:

Given: 2x + y + z = 7 and x - y + 2z = 9.

That means,

a₁ = 2, b₁ = 1, c₁ = 1 and a₂ = 1, b₂ = - 1, c₂ = 2.

To Find: Angle between the planes

Using the above formula for angle θ:

\(\rm \cos \theta = \dfrac{2\times 1+1\times (-1)+1\times 2}{\sqrt{(2^2+1^2+1^2)(1^2+(-1)^2+2^2)}}=\dfrac{3}{\sqrt{6\times6}}=\dfrac{3}{6}=\dfrac{1}{2}\)

⇒ θ = 60°.