Solve 3D Vector Problem: Plane Parallel to ABC Containing D

[EconStudent]

Homework Statement

The four points A = (-1, -1, -1), B = (1, 1, -1), C = (1, -1, 1) and D = (-1, 1, 1) are the vertices of a triangular pyramid. Use vectors to calculate:

d) an equation (in vector form) of the plane parallel to the face ABC and containing the point D

The Attempt at a Solution

I have to admit that I'm at a bit of a loss here. The equation of the plane is given by ax+by+cz=s. If the plane is parallel, then it contains the points A, B, and C.

So would I use the cross product of the vectors ACxAB = ai + bj + ck

Then plug in a, b and c for the equation of the plane, while using the point D for x, y and z, getting:

a(x+1) + b(y-1) + c(z-1) = 0

Working this out, I get:

AB = <2, 2, 0>
AC = <2, 0, 2>
ACxAB= -4i + 4j + 4k

-4(x+1) + 4(y-1) + 4(z - 1) = 0
-4x +4y + 4z = 12
-x + y + z = 3

Any help figuring out if I'm on the right track is greatly appreciated :)

 

[dynamicsolo]

Yep, that's what you do. (The order of the cross product multiplication is unimportant in a problem like this, since if you had done AB x AC , you would just flip the direction of the normal vector to the plane by 180º, which would reverse the sign on all the terms in your plane equation -- that has no effect on the result.)

And you can confirm that D( -1, 1, 1 ) is indeed in the plane -x + y + z = 3 .

 

[micromass]

If the plane is parallel, then it contains the points A, B, and C.

No, not at all. It's parallel to ABC, so in general it won't contain A,B or C.

However, it may be instructive to find the equation of the plane through A,B, and C. Can you find that first?

 

[dynamicsolo]

No, not at all. It's parallel to ABC, so in general it won't contain A,B or C.

However, it may be instructive to find the equation of the plane through A,B, and C. Can you find that first?

I believe EconStudent did misstate that, in that they should say that they want a plane parallel to the one containing A, B, and C. Nonetheless, the calculation for the parallel plane containing D is correct.

 

[micromass]

Ah oops, I didn't read the rest Shame on me...

Yes, what you did is correct. Just ignore my comment.

 

[EconStudent]

I understand my misstatement; my process is correct but my conception was incorrect. It is difficult for me to wrap my head around these multidimensional problems. Thanks for the help.

 

[dynamicsolo]

I understand my misstatement; my process is correct but my conception was incorrect. It is difficult for me to wrap my head around these multidimensional problems. Thanks for the help.

You can picture what the problem is asking for in this way: you have a pyramid with a triangular base (what is called a tetrahedron) and you are finding the equation of a plane parallel to that base which touches the pyramid at its apex (the peak).