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Question

Let pi_ 1 be the xy-plane and pi_ 2 be the yz-plane.
(a) Give a standard equation for pi_1 and for pi_ 2.
(b) Find the angle between pi_1 and pi_2.
(c) Find vectors v parallel to pi_1 and vector w parallel to pi_2 so that the angle between vector v and vector w...
i. 0 ii. 45

a) A vector with only a z-component is perpendicular to the XY plane, eg. (0,0,1). By definition of the location of points on a plane, we have for pi_1

(0,0,1)･(x-x0,y-y0,z-z0) = z-z0 = 0. Since we want the plane to go through the origin, z0 = 0. So pi_1 is given by z = 0. Similarly, pi_2 is x = 0.

b) The angle between the planes is obviously 90°, but to work it out, we want angle between their perpendicular vectors given by the dot product = cos^-1[(0,0,1)･(1,0,0)] = cos^-1(0) = 90°.

c) i. any 2 vectors along the y axis will be parallel, so let V1 = t<0,1,0> and V2 = s<0,1,0>. You can add points y1 and y2 if you want.

ii. For 45°, let V1 = (0,y,0) be a vector in pi_1 and V2 = (0,y,z) be in pi_2 where y = z. The the dot product gives cos^-1[(0,y,0)･(0,y,z)]/|V1||V2|  = cos^-1(y^2/sqrt(y^2 + z^2)) = cos^-1(y^2/sqrt(2y^2)) = cos^-1(1/sqrt(2)) = 45°.

Randy

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#### randy patton

##### Expertise

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

##### Experience

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

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

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

Past/Present Clients
Also an Expert in Oceanography