Will there always be a vector product for 2 vectors in a 3d plane?

[KittiesAre_Cute]

TL;DR

Will there always be a cross product for 2 vectors when they have components in all 3 plane. I mean if they are in xy plane we just take the vector product to be the z axis, but when they have components in all 3 dimensions, will there always be a 2d plane that contains both of them (not the x,y or z plane) so we can take the perpendicular

Hi, it is my first post here so please dont mind any mistakes if you see them. I am currently learning vectors and one thing that I do not understand is if there will always be a vector product for 2 vectors when they have components in all 3 plane. I mean if they are in xy plane we just take the vector product to be the z axis, but when they have components in all 3 dimensions, will there always be a 2d plane that contains both of them (although not the x,y or z plane) so we can take the perpendicular to that plane and get the vector product.

 

[weirdoguy]

Yes.

https://en.m.wikipedia.org/wiki/Cross_product

 

[KittiesAre_Cute]

Yes.

So youre saying 2 vectors in a 3d plane will always have be contained in a common 2d plane?

 

[Orodruin]

Any two (linearly independent) vectors span a two-dimensional subspace. If they are linearly independent the cross product is zero.

 

[KittiesAre_Cute]

Any two (linearly independent) vectors span a two-dimensional subspace. If they are linearly independent the cross product is zero.

Is there a specific reason or proof for that? (Asking so that I can search and learn more)

 

[Ibix]

So youre saying 2 vectors in a 3d plane will always have be contained in a common 2d plane?

Just rotate your coordinate system so that the x axis is parallel to one vector, then rotate it around that vector until the other one lies in the x-y plane.

 

[A.T.]

So youre saying 2 vectors in a 3d plane will always have be contained in a common 2d plane?

At least in one common plane. If the vectors are co-linear or one is a zero-vector, they are contained in infinitely many common planes. Otherwise they span a triangle, which defines a unique plane.

 

[fresh_42]

Is there a specific reason or proof for that? (Asking so that I can search and learn more)

The cross product is a Lie multiplication of a three-dimensional simple real Lie algebra. From here on there are paths in many directions to learn more. Simply start to understand my wording.

If you are more interested in the geometrical part of your question, then look at Geometric Algebra by Eric Chisolm.

If you are especially interested in the cross-product in physics, then search for "right-hand-rule" or "Fleming's rule", e.g. https://www.khanacademy.org/test-pr...es/magnetism-mcat/a/using-the-right-hand-rule

 

[KnightTheConqueror]

I want you to just look around you. Take the corner of your room as origin, and define the x, y and z axes. Now you are in a 3d coordinate space. Now hold one finger of your hand in a direction denoting a vector, and one finger of the other hand denoting another vector in any other random direction. You will realise any two random vectors will form a plane unless they are parallel