Calc III: Vector parallel to the line of intersection

In summary, the conversation involves finding a vector parallel to the line of intersection of the tangent planes at two points on a sphere of radius 1. The solution involves converting spherical coordinates to cartesian coordinates, finding the gradient of the general sphere equation, and setting up equations for the tangent planes at each point. To find the vector, the equations for the tangent planes are set equal to z and then added together to get a single equation of a line. Parametric equations are then used since the question asks for a vector parallel to the line.
  • #1
Seri
6
0

Homework Statement


Hi there,

Two points on the sphere S of radius 1 have spherical coordinates P: \phi = 0.35 \pi, \theta = 0.8 \pi and Q: \phi = 0.7 \pi, \theta = 0.75 \pi. Find a vector parallel to the line of intersection of the tangent planes to S at the points P and Q.


2. The attempt at a solution
First, I converted spherical to cartesian coordinates.
Cartesian:
P (-.72083942, .26684892, -.80901699)
Q (-.5720614, -.41562694, -.70710678)

Then, I found two tangent planes at P and Q. To do this, I found the gradient of the general sphere equation x^2 + y^2 +z^2 = 1. And then the general equation of the tangent plane for each point.

Tangent P = -1.44167x + .533698y - 1.161803z -2.4906521

Tangent Q = -1.1441228x - .83125388y - 1.4142136z -2



3. Relevant equations
I'm lost on what to do after that though. I talked to a tutor about this question and he said to set the two equation so that they equal z. Then add them together to get a single equation of a line. And then use parametrics since it's asking for a vector parallel. Does this sound right?
 
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  • #2
/nudges to the top
Wake up your monday morning with some hearty, delicious calculus! ;)
 
  • #3
Seri said:
Tangent P = -1.44167x + .533698y - 1.161803z -2.4906521

Tangent Q = -1.1441228x - .83125388y - 1.4142136z -2

That looks very complicated. :frown:

The tangent planes at P and Q are perpendicular to OP and to OQ.

So they're simply r.P = 0 and r.Q = 0, and they intersect in the line r.P = r.Q = 0.

Try that! :smile:
 

Related to Calc III: Vector parallel to the line of intersection

What is the definition of a vector parallel to the line of intersection?

A vector parallel to the line of intersection is a vector that lies in the same direction as the line of intersection between two planes. This means that the vector is perpendicular to both planes and has the same direction as the line where the two planes intersect.

How can I determine if a vector is parallel to the line of intersection?

To determine if a vector is parallel to the line of intersection, you can use the cross product of the vector and the normal vectors to the two planes. If the resulting vector is equal to the zero vector, then the original vector is parallel to the line of intersection.

What is the significance of a vector parallel to the line of intersection?

A vector parallel to the line of intersection is significant because it can help determine the relationship between two planes. If a vector is parallel to the line of intersection, it means that the two planes do not intersect and are either parallel or coincident.

Can a vector be parallel to multiple lines of intersection?

Yes, a vector can be parallel to multiple lines of intersection. This can happen when the vector is perpendicular to the planes of intersection for those lines, meaning that the planes are parallel or coincident.

How can I use vectors parallel to the line of intersection in real-world applications?

Vectors parallel to the line of intersection have many real-world applications, such as in engineering and physics. They can be used to determine the direction of forces acting on an object or to calculate the motion of an object in 3D space. They are also used in computer graphics for creating 3D models and animations.

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