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supermiedos
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Homework Statement
find the points on the surface x^2 + y^2 + z^2 = 7 where its tangent plane is parallel to 2x + 4y + 6z = 1
Homework Equations
Equation of a tangent plane:
fx(x - x0) + fy(y - y0) + fz(z - z0) = 0, where fx means partial derivative of f respect to x
n1 X n2 = 0
The Attempt at a Solution
Two planes are parallel if the cross product of their normal vectors is zero. The normal vector of the surface is its gradient, that is: n1 = 2x i + 2y j + 2z k and the normal vector of the plane is
n2 = 2 i + 4 j + 6 k.
when I do n1 X n2 and equal it to zero, i get a system of 3 equations:
12y -8 z = 0, -12x + 4z = 0, 8x - 4y = 0, but it has a infinity number of solutions (y = 2x, z = 3x). what am I doing wrong?
the solutions according to the book is: (1/sqrt(2), sqrt(2), 3/sqrt(2) ) and (-1/sqrt(2), -sqrt(2), -3/sqrt(2)
Thanks in advance