Calculus and Vectors - Determining intersection for lines and planes

[ttpp1124]

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

 

[Mark44]

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.

 

[ttpp1124]

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?

 

[Mark44]

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."

 

[ttpp1124]

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?

 

[Mark44]

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.