I am curious if there is any geometric meaning of cross product between vectors? Is it finding the orthogonal vector?
dnialh
I feel like the inner product is more than how well two vectors line up, because it also heavily depends on how large the input vectors are. The thing that more directly measures how well the vectors line up is <u,v>/|u||v|.
keenan
@Gaming_Hippo Interpreting the cross product as an orthogonal vector works out cleanly only in 3D, where there is a unique line orthogonal to two linearly independent vectors. In dimensions four and greater, you will have higher-dimensional linear subspaces orthogonal to a given pair of independent vectors. E.g., a whole plane of directions in 4D. These ideas can be made more precise using something called exterior algebra, but that goes beyond what we'll study in this course!
I am curious if there is any geometric meaning of cross product between vectors? Is it finding the orthogonal vector?
I feel like the inner product is more than how well two vectors line up, because it also heavily depends on how large the input vectors are. The thing that more directly measures how well the vectors line up is <u,v>/|u||v|.
@Gaming_Hippo Interpreting the cross product as an orthogonal vector works out cleanly only in 3D, where there is a unique line orthogonal to two linearly independent vectors. In dimensions four and greater, you will have higher-dimensional linear subspaces orthogonal to a given pair of independent vectors. E.g., a whole plane of directions in 4D. These ideas can be made more precise using something called exterior algebra, but that goes beyond what we'll study in this course!