Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
1  
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
add comment

3 Answers

up vote 1 down vote accepted

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

share|improve this answer
    
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
add comment

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 $\mathbb{Z}^3$ ?

share|improve this answer
    
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
1  
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
add comment

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.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.