There is a way to project a 3D vector in 4D space, and get its new coordinates?

Question

If yes, I woud like a way to extract it's new coordinates in 4d space (if it is not the same).

for example, a 3D vector (3,2,5), if possible, what would be its coordinates into 4D space?

I thougt about projection, but I dont know if what I asking is possible..

So, sorry if my question sounds dumb.

Thank you!

Answer 1

The most natural way of embedding a $\Bbb R^3$ into $\Bbb R^4$ would be to append a zero to the end of the $3$D vector like so: $(3,2,5)\to(3,2,5,0)$.

Let's envision this as embedding $\Bbb R^2$ to $\Bbb R^3$ of a vector $(3,2)$. If we append a zero to the end of this vector, $(3,2,0)$, we are embedding it into the plane encompassed by $z=0$. If we append a zero in the middle, $(3,0,2)$, we are embedding it into the plane $y=0$, and if append it the beginning, $(0,3,2)$, we are embedding the vector in the plane of $x=0$. We can actually do this with more than just embedding it a plane of zeros with constant planes and dot products, but nevertheless this logic transcribes itself into embedding a vector from $\Bbb R^3$ into $\Bbb R^4$, or really a vector from $\Bbb R^n$ into $\Bbb R^{n+1}$