Determine the angle between the planes 2x - y + 2z = 5 and 3x + 4y = 1.
Answer and explanation
Correct answer: B. arccos(2/15)
- Step 1: Extract normal vectors: n1 = <2, -1, 2> and n2 = <3, 4, 0>.
- Step 2: Calculate dot product: n1 . n2 = 2(3) + (-1)(4) + 2(0) = 6 - 4 + 0 = 2.
- Step 3: Calculate magnitudes: ||n1|| = sqrt(4 + 1 + 4) = 3; ||n2|| = sqrt(9 + 16 + 0) = 5.
- Step 4: Use cosine formula: cos(theta) = |n1 . n2| / (||n1|| ||n2||) = 2 / (3 5) = 2 / 15.
- Step 5: Angle theta = arccos(2/15).
Answer
arccos(2/15).
Why not the others: 'arccos(1/5)' computes 3/15 instead of 2/15; 'arccos(6/25)' squares components incorrectly; 'arccos(2/5)' omits ||n1|| in the denominator.
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