0

In three dimensions, a multivector consists of a scalar, a vector, a bivector and a tri-vector. Is there a term that generalizes these names? For example, in an $n$-dimensional space, can I use the terminology $k$-vector (where $0\leq k\leq n$) to refer to the different scalar and vector types, such that a 0-vector is a scalar, a 1-vector is a vector, a 2-vector is a bivector and a 3-vector is a trivector? I haven't found any such name generalization, but it seems to me like it would be very useful (kind of like how a face of an $n$-polytope can be referred to as a $k$-face, where $0\leq k\leq n$ and $k$ is the dimensionality of the face).

Edit: I found the answer to my own question. They can indeed be called $k$-vectors.

HelloGoodbye
  • 571
  • 2
  • 12
  • 1
    There's also k-blade, for a k-vector that is the product of k perpendicular vector factors. – Peeter Joot Feb 12 '24 at 14:55
  • @PeeterJoot Ok, so a $k$-blade can in some sense be said to be a rank-1 $k$-vector (if you can speak of the rank of a $k$-vector)? For example, if $\mathbb{a}$, $\mathbb{b}$, $\mathbb{c}$ and $\mathbb{d}$ are perpendicular, non-zero vectors, the bivector $\mathbb{ab}$ is a 2-blade (a "biblade"?) because it is the product of $\mathbb{a}$ and $\mathbb{b}$, but $\mathbb{ab}+\mathbb{cd}$ isn't a 2-blade because it can't be formed as the product of two vectors? – HelloGoodbye Feb 13 '24 at 13:31
  • Right, $a b + c d$ would be a 2-vector, but not a 2-blade in that case, but each of $a b$ and $c d$ are 2-blades. – Peeter Joot Feb 28 '24 at 14:54

1 Answers1

0

Scalars, vectors, bivectors and trivectors can indeed collectively be referred to as $k$-vectors.

HelloGoodbye
  • 571
  • 2
  • 12