Line of intersection between two parallel planes?

[sawdee]

How can I find the line of intersection between the planes 2x-y+2x+1=0 and -4x+2y-4x-2=0

I realize these are parallel as they are multiples of each other, but I'm not sure how to solve for the point. I also have to convert this line into parametric, cartesian and vector form.

Sorry for the continuous posts, I have an exam tomorrow and I'm stuck on these questions :/ Can't ask a teacher either.

 

[MarkFL]

When two planes are parallel, you have 2 options:

i) they are the same plane...they then intersect over the entire plane defined by both.

ii) they are distinct...and so they intersect nowhere.

 

[sawdee]

When two planes are parallel, you have 2 options:

i) they are the same plane...they then intersect over the entire plane defined by both.

ii) they are distinct...and so they intersect nowhere.

In terms of that, how can I solve for the question? Is there a way to find this parallel point of intersection?

 

[MarkFL]

In terms of that, how can I solve for the question? Is there a way to find this parallel point of intersection?

The two planes you gave are the same plane, so they "intersect" at infinitely many points...consisting of the entire plane the two equations define.

 

[sawdee]

The two planes you gave are the same plane, so they "intersect" at infinitely many points...consisting of the entire plane the two equations define.

Hmm that does make sense, however the question still asks for the three forms of the line of intersection (parametric, vector and Cartesian). Do i simply write there are infinite solutions, or is there a way to find one?

 

[MarkFL]

Hmm that does make sense, however the question still asks for the three forms of the line of intersection (parametric, vector and Cartesian). Do i simply write there are infinite solutions, or is there a way to find one?

Since the two planes are the same, there are an infinite number of lines within the intersection...there is no one line that is a "best choice." If you are forced to define one line within the given plane, then you could pick any two points within the given plane and construct the line passing through these two points.

Also, you are typing "x" when I think you mean "z"...so let's look at this definition of the given plane:

\(\displaystyle 2x-y+2z+1=0\)

If we let $(y,z)=(0,0)$ we find:

\(\displaystyle x=-\frac{1}{2}\)

And if we let $(x,z)=(0,0)$ we find:

\(\displaystyle y=1\)

And so we have the two points:

\(\displaystyle \left(-\frac{1}{2},0,0\right),\,\left(0,1,0\right)\)

Can you take these two points and describe the line through them?

 

[sawdee]

Since the two planes are the same, there are an infinite number of lines within the intersection...there is no one line that is a "best choice." If you are forced to define one line within the given plane, then you could pick any two points within the given plane and construct the line passing through these two points.

Also, you are typing "x" when I think you mean "z"...so let's look at this definition of the given plane:

\(\displaystyle 2x-y+2z+1=0\)

If we let $(y,z)=(0,0)$ we find:

\(\displaystyle x=-\frac{1}{2}\)

And if we let $(x,z)=(0,0)$ we find:

\(\displaystyle y=1\)

And so we have the two points:

\(\displaystyle \left(-\frac{1}{2},0,0\right),\,\left(0,1,0\right)\)

Can you take these two points and describe the line through them?

Yes sorry, that 4x is supposed to be 4z!

So you mean solve for each component (x,y,z) of the first plane equation and use those 3 points as in the intercepting points?

 

[MarkFL]

Yes sorry, that 4x is supposed to be 4z!

So you mean solve for each component (x,y,z) of the first plane equation and use those 3 points as in the intercepting points?

No, I mean can you find the line passing through the two points I gave?