Show that a line is parallel to and below the plane

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SUMMARY

The discussion focuses on proving that the line defined by the parametric equations x=0, y=t, and z=t is parallel to and below the plane described by the equation 5x - 3y + 3z = 1. The participants establish that the normal vector of the plane, n = (5, -3, 3), is orthogonal to the directional vector of the line, P1P2 = (0, 3, 3), confirming parallelism. To demonstrate that the line is below the plane, they analyze the intersection of the plane with the y-z plane, concluding that the z-values of the line are consistently less than those of the intersection line, L2, which is derived from substituting x=0 into the plane's equation.

PREREQUISITES
  • Understanding of parametric equations in three-dimensional space
  • Knowledge of vector operations, particularly dot products
  • Familiarity with the concept of planes and their equations in 3D
  • Basic graphing skills to visualize lines and planes in the y-z plane
NEXT STEPS
  • Study the properties of dot products and their geometric interpretations
  • Learn how to find the intersection of a plane with coordinate planes
  • Explore graphical methods for visualizing lines and planes in three dimensions
  • Review linear algebra concepts related to vectors and planes
USEFUL FOR

Students and self-learners in mathematics, particularly those studying linear algebra and geometry, as well as educators seeking to clarify concepts related to lines and planes in three-dimensional space.

aero_zeppelin
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Homework Statement



Letting the line be x=0 , y = t and z = t

Show that the line is parallel to and below the plane 5x - 3y + 3z = 1

The Attempt at a Solution



- First I assigned t = 1 and t = 4 to obtain two points of the line:
P1 = (0,1,1) and P2 = (0,4,4)

- Then I created a directional vector P1P2 = (0-0, 4-1 , 4-1) = (0,3,3)

- We know that n = (5, -3, 3)

- So if we prove that n and P1P2 are orthogonal (i.e.their dot product is 0), then THE PLANE AND LINE ARE PARALLEL.

Please tell me if I made any mistake so far...

- But how do you prove the line is BELOW the plane?

Thanks!
 
Last edited:
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... the next question asks for the same but this time ABOVE the plane :P

Thanks in advance!
 
aero_zeppelin said:

Homework Statement



Letting the line be x=0 , y = t and z = t

Show that the line is parallel to and below the plane 5x - 3y + 3z = 1




The Attempt at a Solution



- First I assigned t = 1 and t = 4 to obtain two points of the line:
P1 = (0,1,1) and P2 = (0,4,4)

- Then I created a directional vector P1P2 = (0-0, 4-1 , 4-1) = (0,3,3)

- We know that n = (5, -3, 3)

- So if we prove that n and P1P2 are orthogonal (i.e.their dot product is 0), then THE PLANE AND LINE ARE PARALLEL.

Please tell me if I made any mistake so far...

- But how do you prove the line is BELOW the plane?

Thanks!

Every point on the line has an x-coordinate of 0, and is contained in the y-z plane. Look at the intersection of your given plane with the y-z plane, which will be another line. Call this line L2. To show that the line is below the plane, every point on the line should have a z-value that is less than that of the corresponding point on L2.

Similar idea for your follow-on question.
 
Last edited:
Ok, so I need to find the line of intersection between the given plane and the yz plane.

Hmmm, to do that we set the system:
5x - 3y + 3z = 1
y - z = 0

Right?

Then solve it and obtain:
y = z (or y = t and z = t)
x = 1/5

I don't understand the part about "every z-value must be less than that of the corresponding point on L2."

thx again
 
aero_zeppelin said:
Ok, so I need to find the line of intersection between the given plane and the yz plane.

Hmmm, to do that we set the system:
5x - 3y + 3z = 1
y - z = 0

Right?
No, not at all. The line whose parametric equations you're working with does not intersect the plane, so the system above has no solution.

Every point in the y-z plane (AKA, the plane x = 0) has an x-coordinate of 0, so set x = 0 in the equation of your plane.
aero_zeppelin said:
Then solve it and obtain:
y = z (or y = t and z = t)
x = 1/5

I don't understand the part about "every z-value must be less than that of the corresponding point on L2."
L is the line given parametrically. L2 is the line of intersection between the plane (x - 3y + 3z = 1) and the y-z plane.

Once you figure out the equations of the two lines it should be fairly obvious which one is above, and which one is below.
 
I'm kind of lost now :S I can't see it lol
 
We know the given line and the given plane are parallel - i.e. they DON'T intersect, correct?
 
aero_zeppelin said:
We know the given line and the given plane are parallel - i.e. they DON'T intersect, correct?

Correct.

Your parametric line (L) can be written as z = y, right? (And x = 0.)

What is the equation of line L2, which is the intersection of the plane x - 3y + 3z = 1 in the plane x = 0? Substitute 0 for x in the equation of the plane. What do you get? This really isn't very complicated.
 
yeah, I guess the answer will be obvious. I don't know what I'm missing here loll

Well, if you substitute 0, you get -3y + 3z = 1 , which looks like the equation of a yz plane? but if y = z then you get nothing (0 = 1 ? No solution)
 
  • #10
aero_zeppelin said:
yeah, I guess the answer will be obvious. I don't know what I'm missing here loll

Well, if you substitute 0, you get -3y + 3z = 1 , which looks like the equation of a yz plane?
In the yz plane, this is a line. If you solve for z, what do you get?

aero_zeppelin said:
but if y = z then you get nothing (0 = 1 ? No solution)
The two lines are parallel, so of course they don't intersect. You seem to be forgetting what you're trying to show, that one of them is above the other.
 
  • #11
yeah, I'm getting all mixed up here...

So, solving for z you'd get z = 1/3 + y
 
  • #12
OK, good.

In the x = 0 plane, one line's equation is z = y. The other line's equation is z = y + 1/3. The latter line is in the plane.

Which one is above?

If you're having trouble seeing this, graph both lines. Label the axes with the y-axis being the horizontal axis and the z-axis being vertical axis.

Are you trying to do this without having a drawing?
 
  • #13
Yeah, no drawing (until now), I was trying to get a numerical proof of this.

So, I'm drawing the graph...

It looks like the x=0, y=z line is JUST below the other one (by about 0.3333 units in the z-axis).

I believe this is what you meant by "every z-value must be less than that of the corresponding point on L2"

What confuses me is the fact that we are using the same x=0 and y=z values in the
5x - 3y + 3z = 1 plane ... Are we just trying to set two very similar planes and lines?
 
  • #14
aero_zeppelin said:
Yeah, no drawing (until now), I was trying to get a numerical proof of this.
In this context, "above" and "below" are visual concepts.
aero_zeppelin said:
So, I'm drawing the graph...

It looks like the x=0, y=z line is JUST below the other one (by about 0.3333 units in the z-axis).
How about exactly 1/3.
aero_zeppelin said:
I believe this is what you meant by "every z-value must be less than that of the corresponding point on L2"
Yes.
aero_zeppelin said:
What confuses me is the fact that we are using the same x=0 and y=z values in the
5x - 3y + 3z = 1 plane ... Are we just trying to set two very similar planes and lines?
The title of this thread is "Show that a line is parallel to and below the plane"
The line of the title is the one with the parametric equations. It lies entirely in the plane x = 0 (the y-z plane). "Below" suggests that we look at the z-values on the line and the z-values on the portion of the plane that is directly above the line (IOW, the intersection of the plane with the y-z plane).

You have to work a lot harder to see this if you don't have a drawing.
 
  • #15
It's a lot clearer now! I'm self-teaching all this stuff, so you might guess why I'm having some trouble with it hehe

By the way, do you know any good Linear algebra and vectors book? I've been using Elementary linear algebra (anton/rorre - 9th edition) but I don't think it prepares the student well for this kind of problems

thanks a lot for your help and time
 
  • #16
The book you have is probably OK. I have one by just Anton, 5th Ed. I would guess that your problems stem from some lack of understanding of precalculus concepts. If you're self-teaching, take a look at the materials at khanacademy.org - I think that's the site.
 
  • #17
I'll take a look at that website... thanks for the advice!
 

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