Linear Algebra - Help with Planes

In summary: If not, you can read about it here:https://www.math.niu.edu/~jmathews/LinearAlgebra/GaussianElimination.pdfIn summary, the student is having difficulty understanding concepts in a linear algebra course that are introduced in the first week.
  • #1
36
0

Homework Statement



Consider three planes P1, P2 and P3 where a, b and c are constants.

Plane 1 : x + 2y − z = 5
Plane 2 : ax + y + z = −2
Plane 3 : bx − 2y + cz = 11

1) Find the constants a, b and c such that the three planes intersect in a single point
2) For which values of d and h are the two planes P1 and P3 parallel?

Homework Equations



All I know is there is a single solution when det does not = 0

And the normal vectors to the planes are:
[1, 2, -1]
[a, 1, 1]
[b, -2, c]

The Attempt at a Solution



How do I start this? I really have no clue what to do for part 1. If there's only a single point of intersection, then the determinant can't equal to 0 right? But I'm stuck as to how I can utilize that to help me solve for the constants..

And for part 2, how do you know when two planes are parallel? Is there an equation for it?

Thanks for any help, I've been kinda stumped on this for a while.
 
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  • #2
Er, anybody? Sorry to bump.
 
  • #3
If the planes intersect at a single point then there is a single solution to the augmented matrix, form the augmented matrix and row reduce, you should see the light upon figuring that out.

Edit - if the planes are parallel then that means there is no solution to the system. OR, if they are parallel then they have the same normal vector (or a multiple of it)
 
  • #4
Clever-Name said:
If the planes intersect at a single point then there is a single solution to the augmented matrix, form the augmented matrix and row reduce, you should see the light upon figuring that out.

Edit - if the planes are parallel then that means there is no solution to the system. OR, if they are parallel then they have the same normal vector (or a multiple of it)

Hey thanks for taking the time to answer. The thing is, I JUST got into this course, and everything is still very new to me. The professor hasn't even ever mentioned what an augmented matrix is or what row reduce means. Do you think you can elaborate a bit? Thanks..
 
  • #5
mneox said:
Hey thanks for taking the time to answer. The thing is, I JUST got into this course, and everything is still very new to me. The professor hasn't even ever mentioned what an augmented matrix is or what row reduce means. Do you think you can elaborate a bit? Thanks..

That's odd, because in the linear algebra course I'm taking right now, those concepts are introduced in the first week. Your augmented matrix looks like this:

[tex]\[
\left(
\begin{array}{cccc}
1 & 2 & -1 & 5 \\
a & 1 & 1 & -2 \\
b & -2 & c & 11 \end{array}
\right)\][/tex]

You might have learned row-reduction under the name Gaussian elimination, perhaps.
 

1. What is a plane in linear algebra?

A plane in linear algebra is a two-dimensional flat surface that extends infinitely in all directions. It is represented by a Cartesian equation in three-dimensional space and is defined by a normal vector and a point on the plane.

2. How do you find the equation of a plane in linear algebra?

To find the equation of a plane in linear algebra, you need to know the coordinates of three non-collinear points on the plane or two non-parallel vectors that lie on the plane. You can then use these points or vectors to find the normal vector of the plane, which can be used to form the equation of the plane in Cartesian form.

3. What is the relationship between linear algebra and planes in 3D space?

Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It is closely related to planes in 3D space because planes can be represented as vector spaces and can be transformed using linear transformations.

4. How do you determine if two planes are parallel or perpendicular in linear algebra?

Two planes in linear algebra are parallel if their normal vectors are parallel, which means they have the same direction. Two planes are perpendicular if their normal vectors are perpendicular, which means they are at a 90-degree angle to each other.

5. How is linear algebra used in real-world applications involving planes?

Linear algebra is used in various real-world applications involving planes, such as computer graphics, 3D modeling, and engineering. It is also used in physics to represent and analyze forces and motion in three dimensions, which is essential for understanding the behavior of objects in space.

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