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Question
I am having difficulty figuring out this oproblem. Any help would be greatly appreciated.

What conditions on a, b, c, d will ensure that the plane pi : ax + by + cz = d has each of the following properties?
Write your answers as equations involving only the variables a, b, c, d.

Justify your answers briefly.

(a)   pi is the plane 2x -3y = 1.
(b) pi is orthogonal to the plane x + 5y - 9z = 3.
(c) pi is perpendicular to the line L : (x, y, z) = (1, 4, 2)+ < 2, 3,-1 > t.

Answer
a) For this, this can just read off the coefficients for the scalar eqn for pi: a = 2, b = -3, c = 0, d = 1.

b) If planes are othogonal then their normals are orthogonal: <1,5,-9>・<a,b,c> = 0 ⇒ a + 5b -9c = 0; also d = -3.

c) The normal to pi is parallel to the perpendicular line so that a = 2, b = 3, c = -1.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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