Understanding Planes and Tetrahedrons

[LogarithmLuke]

Hi, I am having trouble understanding why three vectors that lie in the same plane can't form a tetrahedron. If the plane is somewhat vertical or titlted will it not be possible for one vector to higher up than another so that you have a difference in height? Also, for three vectors to form a tetrahedron, can two of the vectors lie in the same plane?

Additionally, if given 4 points, A,B,C,D and you want to check if the points all lie in the same plane. How come it works to check if the crossproduct between vectors AB and AC is perpendicular to vector AD? (i.e finding the crossproduct between vectors AB and AC and then finding the dot product between that and vector AD and seeing if it equals zero).

Any help is appreciated. Please don't hesitate to ask if you need me to clear up on something regarding the questions.

 

[mfb]

Try to draw a proper tetrahedron on paper. It does not matter how you hold that paper in your room: it does not work.

If your third vector is in the plane formed by the other two, the height of your "tetrahedron" is zero, which means you don't have a tetrahedron. This is independent of the orientation of this plane in space.

How come it works to check if the crossproduct between vectors AB and AC is perpendicular to vector AD?

What do you know about the cross product of AB and AC? How is that property, and vectors orthogonal to this cross product, related to the previous part of your post?

 

[LogarithmLuke]

What do you know about the cross product of AB and AC? How is that property, and vectors orthogonal to this cross product, related to the previous part of your post?

The cross product of vectors AB and AC must be perpendicular to both AB and AC. So does that mean vector AD is also perpendicular to AB and AC as its perpendicular to the cross product between them? Or is it that the only way for vector AD to be perpendicular to the cross product of vectors AB and AC is if AB, AC and AD all lie in the same plane?

 

[mfb]

So does that mean vector AD is also perpendicular to AB and AC as its perpendicular to the cross product between them?

No. Apart from its length, the cross product of AB and AC is the only vector that is perpendicular to both. If AD is perpendicular to AB and perpendicular to AC, then it has to be parallel to the cross product.

Or is it that the only way for vector AD to be perpendicular to the cross product of vectors AB and AC is if AB, AC and AD all lie in the same plane?

This is correct. The cross product is perpendicular to the plane of AB and AC. Every vector perpendicular to that cross product is in the plane.

 

[LogarithmLuke]

No. Apart from its length, the cross product of AB and AC is the only vector that is perpendicular to both. If AD is perpendicular to AB and perpendicular to AC, then it has to be parallel to the cross product.
This is correct. The cross product is perpendicular to the plane of AB and AC. Every vector perpendicular to that cross product is in the plane.

I understand it a lot better now, thank you.

Was it true that two of the three vectors forming a tetrahedron can be in the same plane, though?

 

[Mark44]

I understand it a lot better now, thank you.

Was it true that two of the three vectors forming a tetrahedron can be in the same plane, though?

Yes, although this is trivially true. Any two vectors lie in the same plane.

Take one of the two vectors, and move it so that its tail coincides with the tail of the other vector. If the two vectors are parallel (or antiparallel -- i.e., point in opposite directions), they lie along some line in the plane. This includes the possibility that one of the vectors is the zero vector. (If both vectors are zero vectors, then all you have is a single point in the plane.)

If the two vectors point in different directions, the tails of the two vectors are at one point, and the heads of the vectors are at two other points, giving use three distinct points, and thereby defining a plane.

 

[mfb]

Take one of the two vectors, and move it so that its tail coincides with the tail of the other vector.

Moving vectors doesn't really make sense in mathematics. Vectors do not have a "location".

 

[Mark44]

Moving vectors doesn't really make sense in mathematics. Vectors do not have a "location".

Fine, but I think you are being pedantic. We can call them directed line segments that extend from some initial point to a terminal point. A directed line segment like this is considered equivalent to a vector extending from the origin, if the vector and the directed line segment have the same length and direction.