What is the cosine of the angle between the planes x + y + z + 1 = 0 and 2x - 2y + 2z + 1 = 0 ? - Mathematics MCQ Mathematics Mock Test - 3 2026 | ExamTest.live

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What is the cosine of the angle between the planes x + y + z + 1 = 0 and 2 x - 2 y + 2 z + 1 = 0 ?

Solution & Step-by-step Explanation

The normal vectors are n_{1} = (1, 1, 1) and n_{2} = (2, - 2, 2) .The cosine of the angle θ is: cos θ = \frac{∣ n _{1} \cdot n _{2} ∣}{∣ n _{1} ∣∣ n _{2} ∣} n_{1} \cdot n_{2} = (1) (2) + (1) (- 2) + (1) (2) = 2 - 2 + 2 = 2 ∣ n_{1} ∣ = 1^{2} + 1^{2} + 1^{2} = 3 ∣ n_{2} ∣ = 2^{2} + (- 2)^{2} + 2^{2} = 12 = 23 cos θ = \frac{2}{3 \cdot 2 3} = \frac{2}{2 \times 3} = \frac{1}{3}

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What is the cosine of the angle between the planes x + y + z + 1 = 0 and 2 x - 2 y + 2 z + 1 = 0 ?

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