Motivation and proof behind cross products

[Terrell]

this question is a repost from math stackexchange because that guy worded the question so perfectly the question i really wanted to ask about cross products. *please see image below*

as far i can understand, the formula for the cross product is basically that the idea of a cross product is sort of symmetrical to the idea of taking the determinant of a 3x3 matrix(or volume of a parallelepiped) which involves a vector orthogonal to the plane formed by two vectors. and by setting i=<1,0,0>, j=<0,1,0> and, k=<0,0,1>... we in turn get iC_11 + jC_12 + kC_13 such that C_ij are cofactors. thus, the reason why when we take the magnitude of the orthogonal vector, we get the same numeric value of the area of the parallelogram.

 

[FactChecker]

The cross product gives more than the area of the parallelogram. It also gives the orientation of one vector verses the other and the orientation of the parallelogram within the higher dimensional space. The orientation of vector A x B is the opposite of the orientation of vector B x A.

 

[Terrell]

The cross product gives more than the area of the parallelogram. It also gives the orientation of one vector verses the other and the orientation of the parallelogram within the higher dimensional space. The orientation of vector A x B is the opposite of the orientation of vector B x A.

why do we have to define the cross product as something orthogonal to the plane. why not just a value for magnitude? and flipping of signs to indicate orientation...?

 

[FactChecker]

why do we have to define the cross product as something orthogonal to the plane. why not just a value for magnitude? and flipping of signs to indicate orientation...?

Then you wouldn't know the orientation of the plane in 3-space. It's nice to have a vector that can be 'dotted' with a third vector (not in the plane) to calculate the volume of a parallelepiped.

 

[PeroK]

why do we have to define the cross product as something orthogonal to the plane. why not just a value for magnitude? and flipping of signs to indicate orientation...?

Angular momentum, for example, is a vector quantity and follows vector addition. If the angular momenta of two particles were scalars relative to different planes, there would be no way to add them. You can get away with angular momentum as a (signed) scalar as long as all vectors are in a common plane. In the general case, however, you need the full vector representation.

 

[Terrell]

thank you all!