Question
Let a plane pass through the point (1, –1, 2) and is
perpendicular to the vector 2i + j − 2k. Then its Cartesian equation is:Solution
We are given: • A point through which the plane passes: P(1, –1, 2) • A normal vector to the plane: n = 2i + j − 2k → this corresponds to vector n = (2, 1, –2) We are to find the Cartesian equation of the plane. General form of the plane equation: If a plane passes through point (x₀, y₀, z₀) and has normal vector (A, B, C), then the equation is: A(x – x₀) + B(y – y₀) + C(z – z₀) = 0 Substitute: • (x₀, y₀, z₀) = (1, –1, 2) • (A, B, C) = (2, 1, –2) So: 2(x – 1) + 1(y + 1) – 2(z – 2) = 0 Now simplify: 2x – 2 + y + 1 – 2z + 4 = 0 ⇒ 2x + y – 2z + 3 = 0
What will come in place of the question mark (?) in the following series?
1528, 1416, 1204, ?, 480, -32
143 119 107 101 98 ?
96 89 ? 90 94 91
...What value should come in the place of (?) in the following number series?
19, 235, 260, 324, 333, ?
What will come in place of the question mark (?) in the following series?
305...
In each of the following number series, one term is missing. Find the missing term.
3, 5, 20, 24, 144, 150, ?
1029, 1030, 1039, 1064, ?, 1194
75, 80, 95, ?, 155, 200
What will come in place of the question mark (?) in the following series?
1515, 184, ?, 455, 967, 624
7 29 61 ? 211 349
...
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