Solving for Points on a Sphere w/ Parallel Plane

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SUMMARY

The discussion focuses on finding points on the sphere defined by the equation x² + y² + z² = 1 where the tangent plane is parallel to the plane given by 2x + y - 3z = 2. The key approach involves calculating the first-order partial derivatives and using the tangent plane equation z - z0 = fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0). Participants emphasize the importance of identifying the normal vector to the plane and recognizing that any vector from the origin through the intersection point will be orthogonal to the sphere's surface.

PREREQUISITES
  • Understanding of tangent planes in multivariable calculus
  • Knowledge of partial derivatives and their applications
  • Familiarity with the equation of a sphere
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the derivation and application of the tangent plane equation in multivariable calculus
  • Learn how to compute first-order partial derivatives for functions of multiple variables
  • Explore the geometric interpretation of normals to surfaces
  • Investigate the relationship between vectors and surfaces in three-dimensional space
USEFUL FOR

Students and educators in calculus, particularly those focusing on multivariable functions, as well as anyone involved in geometric analysis and algebraic manipulation of equations related to spheres and planes.

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Homework Statement



Find the points on the sphere x2 + y2 + z2 = 1 where the tangent plane is parallel to the plane 2x + y - 3z = 2.

Homework Equations



z - z0 = fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0))

That's all I believe

The Attempt at a Solution



This chapter in our text does not require the use of vectors (eg dot and cross product), so I need to do this strictly algebraically. I found both first order partials, but didn't really know what to do after that. I plug in these partials to the expression above, but I don't think it's possible to solve for x0 and y0 explicitly.

If someone could guide me in the right direction, it would be much appreciated. Thank you all in advance!
 
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first find the normal to the plane

then note any vector from the origin through, will be orthogonal to the sphere face where it intersects it
 

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