Calculus and Vectors - Determining intersection for lines and planes

In summary: The first and second planes intersect in a line, as do the second and third planes. So, you need to subtract the two vectors to get the correct answer.No, neither is correct. Subtracting the two vectors is the wrong way to go.First off, how do the first and second planes intersect? You need to describe this intersection in geometric terms; i.e., in a sentence with words. A sketch of the two planes and how they meet would be helpful. You don't need to include coordinate axes, but you should include a normal vector for each plane.
  • #1
ttpp1124
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4
Homework Statement
Use normal vectors to determine the intersection, if any, for each of the following groups of three planes. Give a geometric interpretation in each case and the number of solutions for the corresponding linear system of equations. If the planes intersect in a line, determine a vector equation of the line. If the planes intersect in a point, determine the coordinates of the point.

I solved it, can someone tell me if I'm correct? Thanks!
Relevant Equations
n/a
IMG_3637.jpg
 
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  • #2
ttpp1124 said:
Homework Statement:: Use normal vectors to determine the intersection, if any, for each of the following groups of three planes. Give a geometric interpretation in each case and the number of solutions for the corresponding linear system of equations. If the planes intersect in a line, determine a vector equation of the line. If the planes intersect in a point, determine the coordinates of the point.

I solved it, can someone tell me if I'm correct? Thanks!
Relevant Equations:: n/a

View attachment 260149
You're correct in saying that the first and third plane are parallel, so won't intersect, but there's a lot more that you didn't address. The problem asks that you "Use normal vectors to determine the intersection, if any, for each of the following groups of three planes."
The first and second planes intersect, as do the second and third planes.
What is the geometric interpretation of the intersections of the first and second planes, as well as of the second and third planes?

How many solutions are there for the two cases I listed?

Be sure you understand all of what the problem is asking for, only part of which I listed above.
 
  • #3
Mark44 said:
You're correct in saying that the first and third plane are parallel, so won't intersect, but there's a lot more that you didn't address. The problem asks that you "Use normal vectors to determine the intersection, if any, for each of the following groups of three planes."
The first and second planes intersect, as do the second and third planes.
What is the geometric interpretation of the intersections of the first and second planes, as well as of the second and third planes?

How many solutions are there for the two cases I listed?

Be sure you understand all of what the problem is asking for, only part of which I listed above.
what does "geometric interpretation" mean?
 
  • #4
ttpp1124 said:
what does "geometric interpretation" mean?
Here's an example. Consider the equations 2x + 3y + z = 5 and x - 2y - z = 3, in space. Normal vectors are, respectively, <2, 3, 1> and <1, -2, -1>. It's obvious that the normals aren't parallel (or antiparallel), so these planes intersect. When two planes intersect, they do so in a line.
Describing the geometry that the equations represent is what is meant by "geometric interpretation."
 
  • #5
Mark44 said:
Here's an example. Consider the equations 2x + 3y + z = 5 and x - 2y - z = 3, in space. Normal vectors are, respectively, <2, 3, 1> and <1, -2, -1>. It's obvious that the normals aren't parallel (or antiparallel), so these planes intersect. When two planes intersect, they do so in a line.
Describing the geometry that the equations represent is what is meant by "geometric interpretation."
I end up with two vector equations, am I correct?
 

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  • #6
ttpp1124 said:
I end up with two vector equations, am I correct?
No, neither is correct. Subtracting the two vectors is the wrong way to go.
First off, how do the first and second planes intersect? You need to describe this intersection in geometric terms; i.e., in a sentence with words. A sketch of the two planes and how they meet would be helpful. You don't need to include coordinate axes, but you should include a normal vector for each plane.

Here's an example of what I'm talking about. The geometric description should say in words, not equations, what the picture is showing -- how the planes are intersecting, and what sort of geometric object the intersection represents.
Planes.png
 

1. What is the purpose of determining intersections for lines and planes in calculus and vectors?

Determining intersections for lines and planes is important in calculus and vectors because it allows us to find the points where two or more lines or planes intersect. This is useful in solving real-world problems, such as finding the point of collision between two moving objects or determining the intersection of a plane and a surface.

2. How do you determine the intersection of two lines in calculus and vectors?

To determine the intersection of two lines, we can use the method of substitution or elimination. In the method of substitution, we solve for one variable in terms of the other in one of the equations, and then substitute that value into the other equation. In the method of elimination, we manipulate the equations so that when they are added or subtracted, one of the variables is eliminated, leaving us with a single variable equation to solve.

3. What is the process for finding the intersection of a line and a plane in calculus and vectors?

The process for finding the intersection of a line and a plane involves finding the point where the line and the plane intersect. This can be done by setting the equations of the line and the plane equal to each other and solving for the variables. The resulting solution will be the coordinates of the point of intersection.

4. Can two planes intersect in more than one point?

No, two planes can only intersect in one point. This is because two planes are defined by three points, and if they intersect in more than one point, they would have to share at least two of those points, making them the same plane.

5. How is the intersection of lines and planes used in real-world applications?

The intersection of lines and planes is used in various real-world applications, such as in engineering, physics, and computer graphics. For example, in engineering, it can be used to determine the point of intersection between two roads or the intersection of a beam and a support structure. In physics, it can be used to calculate the point of collision between two objects. In computer graphics, it is used to create 3D models and determine the perspective of objects in a scene.

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