Find the point of intersection between the line r = (1, -1, 2) + t(2, 1, -1) and the plane 3x + 2*y - z = 8.
Answer and explanation
Correct answer: B. (3, 0, 1)
- Step 1: Express the line in parametric coordinates: x = 1 + 2*t, y = -1 + t, z = 2 - t.
- Step 2: Substitute into the plane equation: 3(1 + 2t) + 2*(-1 + t) - (2 - t) = 8.
- Step 3: Simplify: 3 + 6t - 2 + 2t - 2 + t = 8, which yields 9t - 1 = 8, so 9t = 9 and t = 1.
- Step 4: Substitute t = 1: x = 1 + 2(1) = 3, y = -1 + 1 = 0, z = 2 - 1 = 1.
Answer
(3, 0, 1).
Why not the others: (1, -1, 2) corresponds to t = 0 (gives 3(1) + 2(-1) - 2 = -1 != 8); (5, 1, 0) corresponds to t = 2 (gives 15 + 2 - 0 = 17 != 8); (-1, -2, 3) corresponds to t = -1 (gives -3 - 4 - 3 = -10 != 8).
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