Regarding a geometrical point of view, is there any differences between a vector with n dimensions with a vector with n+1 dimensions whose first n elements are identical to elements of the first vector and its n+1 element is zero?
2026-03-30 12:13:34.1774872814
The differences between a vector with $n$ dimensions with a vector with $n+1$ dimensions whose $n+1$ element is zero?
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The difference is not in the vectors themselves, but in what one can do with the vectors. You cannot add a vector in $n$ dimensions to one in $n+1$ dimensions, but of course you can add two vectors both in $n+1$ dimensions, even if the components of them happen to have value $0$. Likewise, there are other operations that can only be implemented if two vectors "live" in the same-dimensional space, including cross product.