Parallel planes and intersecting planes

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June 8th 2010, 07:44 AM
Kakariki

Parallel planes and intersecting planes

Hey! I am having trouble with a question, and I am hoping one of you can help me with it!

Question
For what values of k will the planes 2x - 6y + 4z + 3 = 0 and 3x - 9y + 6z + k = 0
i) not intersect?
ii) intersect in a line?
iii) intersect in a plane?

Solution
i) for them to not intersect the planes must be parallel. If I multiply plane (1) by 1.5 I get the values in plane (2). Therefore if I set the k value to something that s not a scalar quality of the C value in (1) the planes will be parallel. So:
3x - 9y + 6z + 4 = 0

ii) I believe these two planes to be parallel, so by this logic these two planes can never meet. Therefore no k value will make these planes intersect in a line.
*Unsure about this

iii) I don't understand what the question means by "intersect in a plane". If it means all values of the planes are the same, then I would set the k value to a scalar value of k in the first question. So:
1.5x3 = 4.5
3x - 9y + 6z + 4.5 = 0

I have no idea if I am correct in my thinking here or not. I hope someone here can help me in these solutions!

Thanks!
June 8th 2010, 08:19 AM
Warrenx

Quote:

Originally Posted by Kakariki

Hey! I am having trouble with a question, and I am hoping one of you can help me with it!

Question
For what values of k will the planes 2x - 6y + 4z + 3 = 0 and 3x - 9y + 6z + k = 0
i) not intersect?
ii) intersect in a line?
iii) intersect in a plane?

Solution
i) for them to not intersect the planes must be parallel. If I multiply plane (1) by 1.5 I get the values in plane (2). Therefore if I set the k value to something that s not a scalar quality of the C value in (1) the planes will be parallel. So:
3x - 9y + 6z + 4 = 0

ii) I believe these two planes to be parallel, so by this logic these two planes can never meet. Therefore no k value will make these planes intersect in a line.
*Unsure about this

iii) I don't understand what the question means by "intersect in a plane". If it means all values of the planes are the same, then I would set the k value to a scalar value of k in the first question. So:
1.5x3 = 4.5
3x - 9y + 6z + 4.5 = 0

I have no idea if I am correct in my thinking here or not. I hope someone here can help me in these solutions!

Thanks!

i) Their scalar doesn't matter, they both already are parallel, so it is just their level, and that is at k=/=3.
ii) undefined
iii) They intersect on a plane at k=3, by logic, a plane cannot be skewed unless it is bounded, these are unbounded, so the only alternative is that that k=3, there their cross product = 0.
June 8th 2010, 08:25 AM
undefined

Quote:

Originally Posted by Warrenx

i) Their scalar doesn't matter, they both already are parallel, so it is just their level, and that is at k=/=3.

Shouldn't it be $k\ne\frac{9}{2}$ ?

And likewise for part (iii), $k=\frac{9}{2}$

Also, for part (ii), I would say no solutions (like the OP) rather than undefined. (For a little more info, I agree with quasi's assessment on this page.)