Question

Vectors: Matrices question:
x+y-z=5
2x+2y-4z=6
x+y-2z=3
I realized that normal 2 and 3 are scalar multiples as well as the d-values, therefore, plane 2 and 3 are parallel and coincident. Also, plane 1 must intersect in the form a line. However, when I do my matrics to find the equation of the line, i get this:

Answer 1

1, 1, 0 =3
0, 0, 1= 2
0, 0, 0= 0

Answer 2

So x+y=3
z=2
0z=0

Answer 3

I know that I must create a parameter, but if i let y=t then, i get no y-value.

Answer 4

I get:
1, 1, 0 | 7
0, 0, 1 | 2
0, 0, 0 | 0

\[x + y = 7 \rightarrow x = 7 - y\]\[z = 2\]
rank = 2 and columns = 3 => solution is a line.
Set y = t
\[x = 7 - t\]\[y =0 + t\]\[z = 2 + 0t\]
\[\left(\begin{matrix}x \\ y \\z \end{matrix}\right) = \left(\begin{matrix}7 \\ 0 \\2 \end{matrix}\right) + t\left(\begin{matrix}-1 \\ 1 \\0 \end{matrix}\right)\]

Answer 5

Thanks, i'll check my matrics over. But that is the answer at the back of my textbook. So, thanks! :)

Answer 6

So, if i let y=t, that would be the equation for y?

Answer 7

Yes. It's that simple :)

Answer 8

And replace all the ys with t in the other equations.

Answer 9

Alright, that makes sense. I also have another question to this, but all the planes are parallel and conicident, so I was wondering, in my matrices would I just need one row to equal 0 or 2?

Answer 10

related*

Answer 11

I'm not sure what your question is, but if you have two coincident planes one of the equations always become 0 = 0 when you eliminate, because the vectors spanning one of the planes is a linear combination of the vectors in the other one.

In the case above, the third equation is NOT parallel with the other two because the solution is a line meaning the planes intersect.

Answer 12

Well, what I meant to say was if I was given another question with different planes, and I came to the conclusion that all of theplanes are parallel because of their normals are scalar multiples of each other, and their D values are also scalar multiples, then that would mean that the planes are parallel and concident, so there are infinite solutions. So how would I go about proving that in my matrices? Would it be suffice having one row of the matrices =0? or 2 rows of the matrices=0? orrrrrrrrr, do i even need to do the matrices, since there are an infinite amount of solutions. But, i think i need to do the matrices because my teacher wants me to, so.. :P

Answer 13

aah okay! :)

To prove the planes are parallel or coincident, you (like you wrote) have to prove they are scalar mutiplications of each other, and this is done by solving the equation system they represent.
0 = 0 means the planes coincident
0 = some constant no equal to zero, means the planes are parallel.

You don't have to solve using a matrix to but it's convenient and good practice :)

Another way to prove that one or more of the planes are parallel or convenient is show that the determinant is equal to 0 (if A is square matrix).

Ab = x, A is a n x n and det(A) = 0
=> The equation have no or an infinite amount of solutions => One ore more of the planes are parallel or convenient.

Answer 14

parallel or coincident*

Answer 15

Merci beaucoup. :)

Answer 16

Yw :)