What does the geometric representation of a vector cross product imply?

Question

I recently began studying vectors. I read about the vector cross product operation and came across a geometrical representation of the same.. It said ( with proof) that the cross product indicates the area of the parallelogram contained between the multiplied vectors. Does that mean vectors that lie in the specific area or something? Thanks

Answer 1

No. The resulting vector is perpendicular to the factor vectors. However its length is the size of the area of the parallelogram.

$$ \lVert a \times b \rVert = \lVert a \rVert \, \lVert b \rVert \sin \angle(a, b) $$

Answer 2

Not ar all. It could not mean that, since the cross product of two vectors $v$ and $w$ is orthogonal to both of them. It only means that the length of $v\times w$ is equal to the area of the parallelogram that you mentioned.