Linear Algebra orthogonality problem

In summary: That is the direction vector of the line. Now you just need a point and you can write the equation of the line.Ok so the line is x=-z, but what about y? In summary, the equation for the line of intersection of the two planes x + y + z = 0 and x - y + z = 0 in R3 is x = -z, and the direction vector of the line is (1, 0, -1). The equation can be written as (x-0)/1 = (y-0)/0 = (z-0)/-1, and a point on the line can be chosen to complete the equation.
  • #1
mpittma1
55
0

Homework Statement


Let W be the intersection of the two planes

x + y + z = 0 and x - y + z = 0

In R3. Find an equation for Wτ


Homework Equations





The Attempt at a Solution



So, W = {(x, y, z) l 2y =0}

I don't think that is a correct was to represent W being the intersection of the planes though.

I can find Wτ after I know how to find my equation for W.

Any thoughts for how to find the equation for W?
 
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  • #2
mpittma1 said:

Homework Statement


Let W be the intersection of the two planes

x + y + z = 0 and x - y + z = 0

In R3. Find an equation for Wτ


Homework Equations





The Attempt at a Solution



So, W = {(x, y, z) l 2y =0}

I don't think that is a correct was to represent W being the intersection of the planes though.

I can find Wτ after I know how to find my equation for W.

Any thoughts for how to find the equation for W?

##y=0## alright, but you need more. You also need ##x=-z## for ##(x,y,z)## to be on both planes. Do you see how to write the equation from that?
 
  • #3
LCKurtz said:
##y=0## alright, but you need more. You also need ##x=-z## for ##(x,y,z)## to be on both planes. Do you see how to write the equation from that?

Im not seeing how to get x = -z...
 
  • #4
mpittma1 said:
Im not seeing how to get x = -z...

Look at the equations of the two planes when ##y=0##.
 
  • #5
LCKurtz said:
Look at the equations of the two planes when ##y=0##.

Ok so you "Let" x = -z, so that way when y=0 the equation for the two planes become x + z = 0

so x has to be equal to - z to make -z + z = 0 right?
 
  • #6
Yes. So what is the equation of the line of intersection?
 
  • #7
x+z = 0?
 
  • #8
No. That is the equation of a plane in 3D. You might look, for example, here:

http://www.math.hmc.edu/calculus/tutorials/linesplanesvectors/
 
  • #9
A tip is to see x=-z as (x-0)/1 = (z-0)/-1.
 

1. What is the concept of orthogonality in linear algebra?

Orthogonality in linear algebra refers to the relationship between two vectors or matrices where they are perpendicular to each other. This means that the angle between the two vectors is 90 degrees, and their dot product is equal to zero.

2. How is orthogonality used in solving linear algebra problems?

Orthogonality is a useful tool in solving linear algebra problems as it allows us to break down a complex problem into simpler ones. It helps in finding the basis and dimensions of a vector space, as well as in solving systems of linear equations and finding the orthogonal projections of a vector onto a subspace.

3. What are some real-world applications of orthogonality in linear algebra?

Orthogonality has many applications in various fields, including computer graphics, signal processing, and robotics. In computer graphics, it is used to create realistic 3D images by calculating the angles between light sources and pixels on a screen. In signal processing, it is used to filter out unwanted noise and extract important signals. In robotics, it is used to control the movement and orientation of robots.

4. How is orthogonality related to other concepts in linear algebra?

Orthogonality is closely related to other concepts in linear algebra, such as linear independence, inner product spaces, and orthonormal bases. It is also used in conjunction with other concepts, such as eigenvectors and singular value decomposition, to solve various problems in linear algebra.

5. Can orthogonality be extended to higher dimensions in linear algebra?

Yes, orthogonality can be extended to higher dimensions in linear algebra. In two dimensions, two vectors are considered orthogonal if their dot product is equal to zero. In three dimensions, three vectors are considered orthogonal if their dot products with each other are equal to zero, and so on for higher dimensions. Orthogonality is a fundamental concept in linear algebra and is applicable to vectors and matrices of any dimension.

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