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What do I call an arbitrary element of this set of vectors? $$ \begin{align*} \{&\langle 1, 0, 0 \rangle, \\ &\langle 0, 1, 0 \rangle, \\ &\langle 0, 0, 1 \rangle, \\ &\langle -1, 0, 0 \rangle, \\ &\langle 0, -1, 0 \rangle, \\ &\langle 0, 0, -1 \rangle \} \\ \end{align*} $$

The significance is that this set contains every unit vector which lies on a cubical grid (is in $\mathbb{Z}^3$, as are all sums of elements). In particular, they are all possible directions of motion to adjacent grid points.

It differs from the standard basis for $\mathbb{R}^3$ in including the inverse of each basis vector.

The context is computer game/graphics programming.

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The axis-aligned unit vectors? – Rahul Jan 17 '12 at 0:13
Rahul Narain: Ding. Perfect. Make that an answer, please? – Kevin Reid Jan 17 '12 at 1:53
up vote 1 down vote accepted

I'd call them the axis-aligned unit vectors.

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This is especially good as “axis-aligned” is a common term in the field (“axis-aligned bounding boxes”). – Kevin Reid Jan 17 '12 at 2:06

I don't have the rep to just comment but is there any reason why you couldn't just call them the unit vectors in $\mathbb{Z}^3$ ?

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Here’s a reason: $\mathbb{Z}^3$ isn’t a vector space, since you can’t define scalar multiplication on it. I also think that doesn't flow well in the sort of context it will be used in, e.g. “// Return a rotation code to bring the +z vector to match the given unit vector in Z^3 – Kevin Reid Jan 17 '12 at 1:56
That's true. $\mathbb{Z}^3$ is a well defined lattice in $\mathbb{R}^3$ so is relatively unambiguous as far as terminology goes though perhaps it may be of the wrong 'type' for computational purposes – dke Jan 17 '12 at 2:09

Typically we name the unit vectors $\hat{i}=\begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}$, $\hat{j}=\begin{pmatrix} 0 \\ 1 \\ 0\end{pmatrix}$, $\hat{k}=\begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}$.

As for $\begin{pmatrix} -1 \\ 0 \\ 0\end{pmatrix}$ we simply call it $-\hat{i}$. The same applys for the remaining two vectors.

For further reading you can visit here.

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I am looking for a single name which applies to any element of this set, not names for each of the six. – Kevin Reid Jan 17 '12 at 0:05

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