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blsea [12.9K]
2 years ago
11

Find the cosine of the angle between the planes x+y+z=0 and 4x+3y+z=1. Answer: cosθ=

Mathematics
1 answer:
marta [7]2 years ago
4 0

The angle between the planes is the same as the angle between their normal vectors, which are

<em>n</em><em>₁</em> = ⟨1, 1, 1⟩

<em>n</em><em>₂</em> = ⟨4, 3, 1⟩

The angle <em>θ</em> between the vectors is such that

⟨1, 1, 1⟩ • ⟨4, 3, 1⟩ = ||⟨1, 1, 1⟩|| ||⟨4, 3, 1⟩|| cos(<em>θ</em>)

Solve for cos(<em>θ</em>) :

4 + 3 + 1 = √(1² + 1² + 1²) √(4² + 3² + 1²) cos(<em>θ</em>)

8 = √3 √26 cos(<em>θ</em>)

cos(<em>θ</em>) = 8/√78

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