(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F(x, y, z) = 0 and G(x, y, z) = 0 are orthogonal at a point P where ∇F ≠ 0 and ∇G ≠ 0 if and only if FxGx + FyGy + FzGz = 0 atP(b) Use part (a) to show that the surfaces z2 = x2 + y2 and x2 + y2 + z2 = r2 are orthogonal at every point of intersection. Can you see why this is true without using calculus?

(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F(x, y, z) = 0 and G(x, y, z) = 0 are orthogonal at a point P where ∇F ≠ 0 and ∇G ≠ 0 if and only if FxGx + FyGy + FzGz = 0 atP(b) Use part (a) to show that the surfaces z2 = x2 + y2 and x2 + y2 + z2 = r2 are orthogonal at every point of intersection. Can you see why this is true without using calculus?

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.6: The Three-dimensional Coordinate System

Problem 41E: Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?

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Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?
(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F( x, y, z) = 0 and G(x,y, z) = 0 are orthogonal at a point P where ∇F≠ 0 and ∇F≠ 0 if and only if FxGx +FyGy+FzGz=0 at P (b) Use part (a) to show that the surfaces z2 = x2 +y2 and x2 +y2 + z2= 12are orthogonal at every point of intersection. Can you see why this is true without using calculus?
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