1

Let $a\in\mathbb R^3\setminus\left\{0\right\}$. I'm a bit rusty at this basic geometry stuff, so please bear with me. I want to find an orthonormal basis $(u,v,w)$ of $\mathbb R$ such that $a/|a|$ has coordinates $(0,0,1)$. Gram-Schmidt will do the job, I know, but I want to construct $(u,v,w)$ such that it forms a (i) right-handed and (ii) left-handed system.

We clearly need $w:=a/|a|$. Now I guess we may pick any $b\in\mathbb R$ which is not parallel to $w$ (my idea is to let $i:=\operatorname{arg min}_j|w_j|$ and $b:=e_i$ ($i$th standard unit vector; is this a good idea or are there any issues?) and let $$u:=\frac{b\times w}{|b\times w|}.$$ $u$ is obviously perpendicular to $u$, but we could likewise taken $$\tilde u:=\frac{w\times b}{|w\times b|}=-u.$$

Does it matter whether we pick $u$ or $\tilde u$? I think it shouldn't and I think that the handedness now depends only on the choice of $v$. So, do we need to choose $w\times u$ or $u\times w$ (or $w\times\tilde u$ or $w\times\tilde u$)?

0xbadf00d
  • 13,422

1 Answers1

1

It doesn't matter. Suppose that you took $u$. Then let $v=u\times\frac a{\lVert a\rVert}$. Then the basis $\left\{u,\frac a{\lVert a\rVert},v\right\}$ is an orthogonal basis with the same orientation as the canonical basis, and therefore $\left\{v,u,\frac a{\lVert a\rVert}\right\}$ is also an orthogonal basis with the same orientation as the canonical basis.

José Carlos Santos
  • 427,504
  • The actual question is: How do we need to define $(u,v,w)$ with $w=a/|a|$ such that it is a (i) right-handed and (ii) left-handed system? – 0xbadf00d Jul 25 '19 at 16:54
  • I know. And right-handed means that it has the same orientation as the canonical basis, whereas left-handed means that it has the opposite one. – José Carlos Santos Jul 25 '19 at 17:01
  • (a) Yes, that's clear to me. Okay, is $(u,v,w)$ (as defined in the question, but with $v=w\times u$) form a right-handed system? If so, why does it have the same orientation as the canonical basis? (b) Can we replace $u$ with $\tilde u$ in (a)? (c) How do we obtain the left-handed system? – 0xbadf00d Jul 25 '19 at 17:09
  • (a) Yes, it forms a right-handed system. That's a property of the cross-product. And “forms a right-handed system” is equivalent to “it has the same orientation as the canonical basis”. (b) Yes. (c) You take $\left{-v,u,\frac a{\lVert a\rVert}\right}$. – José Carlos Santos Jul 25 '19 at 17:13
  • Where can I find a proof of this property? And what exactly is this property? Maybe $(a\times b,b,b\times c)$ is always right-handed? And I guess the orientation is preserved if and only if the basis change is actually a rotation, i.e. $(e_1,e_2,e_3)\to(e_3,e_1,e_2)\to(e_2,e_3,e_1)$. – 0xbadf00d Jul 25 '19 at 17:15
  • Here, for instance. – José Carlos Santos Jul 25 '19 at 17:25
  • Thank you for the reference. Is there any better advice for the choice of $b$ than the one that I gave? – 0xbadf00d Jul 25 '19 at 17:53
  • 1
    I think that it is a good choice. – José Carlos Santos Jul 25 '19 at 17:58
  • Is thre a reason you've changed the sign of $v$ for the left-handed system? If I got things right, we call $(u,v,w)$ right-handed resp. left-handed if $\det(u,v,w)>0$ resp. $\det(u,v,w)<0$. Now interchanging an uneven number of columns canges the sign of the determinant, so we should have $\det(u,v,w)=-\det(v,u,w)$. Sinc you agree on that $(u,v,w)$ from the question, with $v=w\times u$, is right-handed, $(v,u,w)$ should be left-handed. Am I missing something? – 0xbadf00d Jul 25 '19 at 18:36
  • What I wrote in my answer was that $\left(v,u,\frac a{\lVert a\rVert}\right)$ is a right-handed system with respect to which the coordinates of $\frac a{\lVert a\rVert}$ are $0$, $0$; and $1$. So, $\left(-v,u,\frac a{\lVert a\rVert}\right)$ is a left-handed system with respect to which the coordinates of $\frac a{\lVert a\rVert}$ are also $0$, $0$; and $1$. – José Carlos Santos Jul 25 '19 at 18:44
  • Sure, but the same would apply to $(v,u,w)=(v,u,a/|a|)$ wouldn't it? – 0xbadf00d Jul 25 '19 at 18:59
  • I am sorry, but I do not understand your question. – José Carlos Santos Jul 25 '19 at 19:01
  • $(v,u,w)$ with $u,w$ as in the question and $v:=w\times u$ is left-handed and the coordnate of $a$ wrt that system is $(0,0,1)$. That's what I mean. Or am I missing something? – 0xbadf00d Jul 25 '19 at 19:03
  • Why would the coordinates of $\frac a{\lVert a\rVert}$ with respect to that basis be $0$, $0$, and $1$? – José Carlos Santos Jul 25 '19 at 19:59