Equation of the tangent line in the direction of a vector

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SUMMARY

The discussion focuses on finding the equation of the tangent line to the surface defined by the function f(x,y) = x + e^(xy) at the point (2,-1) in the direction of the vector u = <1,-2>. The user identifies the need to compute the gradient vector of the function to address the problem effectively. A proposed solution involves determining the intersection of the surface with the vertical plane defined by y = -2x and subsequently finding the tangent to that curve at the specified point. This approach clarifies the relationship between the tangent line and the vector direction.

PREREQUISITES
  • Understanding of gradient vectors in multivariable calculus
  • Familiarity with tangent lines and their equations
  • Knowledge of surface intersections with planes
  • Basic proficiency in exponential functions and their properties
NEXT STEPS
  • Study the computation of gradient vectors for multivariable functions
  • Learn how to find tangent lines to curves defined by implicit functions
  • Explore the concept of surface intersections with planes in three-dimensional space
  • Investigate the projection of vectors onto tangent planes
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Students and professionals in mathematics, particularly those studying calculus and vector analysis, as well as anyone involved in solving problems related to tangent lines and surface interactions.

alex steve
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I am having issues figuring out how to do the "in the direction of the vector" part of my problem

I have found the equation of the tangent line but i do not know how to the the next part.

My question asks:

Find the equation of the tangent line to the surface defined by the function f(x,y) = x + e^(xy) at point (2,-1) in the direction of the vector u = <1,-2>

Would i have to figure out the gradient vector of my equation that i find?
 
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The phrasing of the question is a little obscure, because the vector u is a 2D vector in a plane that is not tangent to the surface.
I think what they mean is that they want the equation of the tangent line to the surface whose projection on the x-y plane is the vector u. Or, what amounts to the same thing, the projection of u on the tangent plane to the surface at the given point.

One way to do that is to find the curved line that is the intersection between the surface and the vertical plane with equation y=-2x. Then find the tangent to that line at the given point.

There may well be a quicker way, but that's all that occurs to me right now.
 

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