Finding Parametric Equations for Tangent Line of Surface Intersection

In summary, the student is finding parametric equations for the tangent line of the curve of intersection between two surfaces at a given point, but is struggling with evaluating the determinant for the cross product of two gradients. They are also reminded that there is not a unique tangent vector and can be multiplied by a constant.
  • #1
sapiental
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Homework Statement




5. Find parametric equations for the tangent line to the curve of intersection of the surfaces
z^2 = x^2 + y^2 and x^2 + 2y^2 + z^2 = 66 at the point (3, 4, 5).


The Attempt at a Solution



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

Partial derivz:

f'x = 2x
f'y = 2y
f'z = -2z

g'x = 2x
g'y = 4y
g'z = 2z

grad f = <2x,2y,-2z>
grad g = <2x,4y,2z>


grad f (3,4,5) = < 6,8,-10>
grad g (3,4,5) = <6,16,10>

then v = [(grad f) X (grad g )(cross product)

= |i j k | = 180i + 220j - 48k
|6 8 -10|
|6 16 10|


this approach is wrong because the solutions manual gives me

-10i + 4j - 2k for that vector


please help.

Thanks
 
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  • #2
There is something going wrong with your determinant evaluation. I get 240i-120j+48k. Also remember there is not a single tangent vector - you can always multiply by a constant and still have a tangent vector.
 

1. What are parametric equations?

Parametric equations are a set of equations used to describe the coordinates of points on a curve or surface. They involve expressing the coordinates in terms of one or more parameters, such as time or distance.

2. How do you find the parametric equations for the tangent line of a surface intersection?

To find the parametric equations for the tangent line of a surface intersection, you first need to find the equation of the surface and the point of intersection. Then, you can use the gradient vector of the surface at that point to determine the direction of the tangent line. Finally, plug in the point of intersection and the direction vector into the parametric equation formula to find the equations for the tangent line.

3. Why do we use parametric equations for tangent lines?

We use parametric equations for tangent lines because they allow us to easily determine the coordinates of points on a curve or surface. They also provide a way to find the slope or rate of change at a specific point, which is essential in understanding the behavior of a curve or surface.

4. What is the importance of finding the tangent line of a surface intersection?

Finding the tangent line of a surface intersection is important because it helps us understand the behavior of the surface at that specific point. It also allows us to calculate the slope or rate of change at that point, which is useful in many applications such as optimization and motion analysis.

5. Are there any real-world applications of finding parametric equations for tangent lines of surface intersections?

Yes, there are many real-world applications for finding parametric equations for tangent lines of surface intersections. For example, in physics, these equations can be used to analyze the motion of objects on curved surfaces. In engineering, they can be used to determine the slope of a surface for construction or design purposes. Additionally, in computer graphics, these equations are used to create smooth and realistic 3D models of curved surfaces.

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