find the angle between the two

Question

find the angle between the two surfaces at the given point of intersection.

(compute this angle as the angle between the normals to the surfaces at this point)

(x^2)+(y^2)+(z^2)=4

(z^2)+(x^2)=2

(1 , 根號2 , 1)

Answer 1

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let f(x, y, z) = (x^2)+(y^2)+(z^2)-4

g(x, y, z) = (z^2)+(x^2)-2

gradient f(x, y, z) = (2x, 2y, 2z)

gradient f(1 , 根號2 , 1) = (2, 2根號2, 2)

gradient g(x, y, z) = (2x, 0, 2z)

gradient g(1, 根號2, 1) = (2, 0, 2)

cos(angle) = (2, 2根號2, 2)‧(2, 0, 2) / [│(2, 2根號2, 2)│*│ (2, 0, 2)│]

= 根號2/2

angle = 45度或π/4

2008-03-19 01:23:22 補充：

another angle is 135度 or 3π/4 (supplementary)

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