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yopy
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I looked at the answer key and it said (1/3) (1,-2,1)
Does anyone know where the 1/3 came from?
Last edited:
Calculus 3, dealing with tangent planes and surfaces.
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SUMMARY
The discussion centers on calculating the gradient of an ellipsoid and finding tangent planes using the vector <4, -4, 6>. The user initially derived the point (1, -2, 1) but later discovered that the correct point is (1/3, -2/3, 1/3) due to normalization by dividing by 3. This adjustment ensures that the point lies on the ellipsoid defined by the equation 2x² + y² + 3z² = 1. The necessity of parallel normal vectors for tangent planes is also emphasized.
PREREQUISITES- Understanding of ellipsoids and their equations
- Knowledge of gradients in multivariable calculus
- Familiarity with vector operations and parallelism
- Experience with solving systems of equations
- Study the properties of ellipsoids and their gradients
- Learn how to derive tangent planes from gradients
- Explore vector normalization techniques in calculus
- Investigate the relationship between normal vectors and parallel planes
Students studying multivariable calculus, particularly those focusing on tangent planes and gradients, as well as educators looking for examples of ellipsoid applications in calculus.
I looked at the answer key and it said (1/3) (1,-2,1)
Does anyone know where the 1/3 came from?
- #2
tiny-tim
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Hi yopy! 
You'll only get a proportion out of that.
2*12 +(-2)2 +3*1 = 9,
so you have to divide everything by 3.
yopy said:
You'll only get a proportion out of that.
2*12 +(-2)2 +3*1 = 9,
so you have to divide everything by 3.
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HallsofIvy
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One obvious point should be that (1, -2, 1) is not even on the ellipse! [itex]2(1)^2+ (-2)^2+ 3(1)^2= 9[/itex] not 1!
In order that two planes be parallel, their normal vectors must be parallel, but not necessarily equal. As tiny-tim said, you should have [itex]<4x, 2y, 6z>= \lambda<4, -4, 6>[/itex] for some number [itex]\lambda[/itex]. That together with [itex]2x^2+ y^2+ 3z^2= 1[/itex] gives x, y, and z.
In order that two planes be parallel, their normal vectors must be parallel, but not necessarily equal. As tiny-tim said, you should have [itex]<4x, 2y, 6z>= \lambda<4, -4, 6>[/itex] for some number [itex]\lambda[/itex]. That together with [itex]2x^2+ y^2+ 3z^2= 1[/itex] gives x, y, and z.
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