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

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In summary: 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, And 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) = 0working this out, I get:AB = <2, 2, 0>AC = <2, 0, 2>
  • #1
EconStudent
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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 :)
 
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  • #2
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 .
 
  • #3
EconStudent said:
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?
 
  • #4
micromass said:
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.
 
  • #5
Ah oops, I didn't read the rest :frown: Shame on me...

Yes, what you did is correct. Just ignore my comment.
 
  • #6
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.
 
  • #7
EconStudent said:
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).
 

What is a 3D vector problem?

A 3D vector problem involves finding the magnitude and direction of a vector in three-dimensional space. This can involve solving for unknown values or manipulating vectors using mathematical operations.

What does it mean for a plane to be parallel to ABC?

A plane is parallel to the three-dimensional vectors ABC if it does not intersect those vectors and maintains the same direction and spacing as those vectors.

How do you determine if a plane is parallel to ABC?

To determine if a plane is parallel to ABC, you can use the dot product of the normal vector of the plane and the direction vector of ABC. If the dot product is equal to 0, then the plane is parallel to ABC.

What is the normal vector of a plane?

The normal vector of a plane is a vector that is perpendicular to the plane and points away from it. It is used to determine the orientation and direction of the plane.

What is the significance of point D in a 3D vector problem?

Point D is a point on the plane that is parallel to ABC. It is used to define the position of the plane in three-dimensional space and is essential in solving the vector problem.

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