The angle between the planes x − y + z = 1 and 2x + y

Solution

The angle between two planes is equal to the angle between their normal vectors. Normal vector of plane 1: n₁ = (1, −1, 1) Normal vector of plane 2: n₂ = (2, 1, −3) Use the formula: cos(θ) = |n₁ · n₂| / (||n₁|| · ||n₂||) Dot product: n₁ · n₂ = (1)(2) + (−1)(1) + (1)(−3) = 2 − 1 − 3 = −2 ⇒ |n₁ · n₂| = 2 Magnitude of n₁ = √(1² + (−1)² + 1²) = √3 Magnitude of n₂ = √(2² + 1² + (−3)²) = √14 cos(θ) = 2 / (√3 · √14) = 2 / √42 ⇒ θ = cos⁻¹(2 / √42)