Question

Ok. Now which of the following describes the intersection of x+y+z=3 and x+y+z=6?
a.Intersect at a point
b.at a line
c.dont intersect
d.they are the same plane

Answer 1

So these are planes, OK?
And the normal vectors are parallel, do you know what that means.?

Answer 2

Nope. I have to teach myself this so i am lost about vectors but i know what a parallel is

Answer 3

Well, I don't know how you would know this but these two equations are for parallel planes.

Answer 4

Which mean they don't intersect then since there parallel

Answer 5

Yep.

Answer 6

But how would you know that?

Answer 7

I don't know i guess because you said they were parallel but how did you know that they were

Answer 8

Because their normal vectors are the same. <1,1,1>

Answer 9

normal meaning perpendicular

Answer 10

For both because theirs nothing in-front of the x y or z. And the number at the end just tell you how wide it is. I am guessing

Answer 11

Close enough. They are not really 3 units apart but that's another story. Main thing is they are parallel planes.

Answer 12

So what does the number behind the equal sign means to do.

Answer 13

I would leave that behind for a minute. What do you know about planes and how would you answer this question with no help?

Answer 14

I guess you could do that
two planes will never intersect at a point
two planes that are the same will have the same equation (within a constant)

Answer 15

I know how to find the distance if the x,y is given. But nothing really once they add a z....

Answer 16

Well it seems crazy to me.
My advice: go with what we have, and if you have a new Q about planes, ping me OK?

Answer 17

Ok. Well my next question is.. The graph of -3x+2y=-6z=18 intersects at which three points. Now how do you solve this

Answer 18

intersects what?
and are you sure -3x+2y=-6z=18
??

Answer 19

be back in 10 or 15

Answer 20

ok and i meant -3x+2y-6z=18

Answer 21

Do you mean intersects the coordinate axes at which 3 points?

Answer 22

To find where it intersects the x-axis, set y=z=0 to get
-3x+2(0)-6(0)=18, so x=-6. The intersection point is therefore (-6,0,0).
You can do the same for the two other axes.

Answer 23

so just plug in zero for all of them and ill find my answer

Answer 24

Not all of them, just two at a time and solve for the remaining variable.
Please first confirm that you do want the intersection of the plane with the axes.

Answer 25

ok i get it now

Answer 26

Great!
Is it clear how you would find out if two planes are parallel?

Answer 27

nope

Answer 28

You know how to find the normal vector (like the <1,1,1> that you mentioned)?

Answer 29

Yes because i know that by infront of x is a 1

Answer 30

Basically, that's what it is.
Put the equation of the plane into the form
ax+by+cz=d
the normal vector would be <a,b,c>.
So far so good?

Answer 31

Yes thats the basic

Answer 32

For two given planes, you would find the normal vectors for each one, say
<a,b,c> and <p,q,r>.
Check if p/a=q/b=r/c (assuming they are all non-zero).
If equality holds, then the two planes are parallel.
Example:
Is 2x+3y+4z=6 parallel to -4x-6y-8z=-12?

Answer 33

ok ok i see

Answer 34

So are they parallel?

Answer 35

No but i dont know how to graph it

Answer 36

In fact, they are parallel, because
-4/2 = -6/3 = -8/4 (=-2)
So the two planes are actually parallel.

Answer 37

Ok now i am back lost this is getting kind of confusing

Answer 38

The easy way to see this is compare the two normal vectors,
<2,3,4> and <-4, -6, -8>
If you can multiply one by a constant to get the other, then the two planes are parallel.
Here the constant is -2, because
-2<2,3,4>=<-4,-6,-8> therefore the two planes are parallel.

Is that OK?

Answer 39

Oh makes sense so they have common factor

Answer 40

Kind of!
I would say that if the components are proportional, then they are parallel.
So far so good?

Answer 41

Yes thanks for slowing it down for me

Answer 42

It is important that you understand every step. That's why I am here.
Next:
do you know how to find out if the two planes coincide?
That is if the two planes have infinite points of intersection?

Answer 43

Oh no whats that

Answer 44

If two parallel planes are so close to each other that they actually coincide.
This is a special situation where the intersection of the two planes is not a line any more, but a plane!

Answer 45

so if they are real close then its a coincide but how far part of way

Answer 46

To check if they are coincident, you would multiply one of the equations by the factor k to make the left-hand sides equal. If the right-hand sides also equal, then the two planes coincide, or are identical.
Take the same two planes we had:
2x+3y+4z=6 ...(1)
-4x-6y-8z=-12 ....(2)

Multiplying plane (1) by -2, we get
-4x-6y-8z=-12 ...(3)

The left hand side of (3) is the same as plane 2, so (1) and (2) are parallel (as before). The right-hand side of -12 is also equal to the right-hand side of (2), so (1) and (2) are coincident, and the intersection is a plane, namely plane (1) or plane (2).

Answer 47

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