# Coordinate free proof for $a\times (b\times c) = b(a\cdot c) - c(a\cdot b)$

The vector triple product is defined as $\mathbf{a}\times (\mathbf{b}\times \mathbf{c})$. This is often re-written in the following way: \begin{align*}\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})\end{align*} This is a very useful identity for integrating over vector fields and so on (usually in physics).

Every proof I have encountered splits the vectors into components first. This is understandable, because the cross product is purely a three dimensional construct. However, I'm curious as to whether or not there is a coordinate free proof of this identity. Although I don't know much differential geometry, I feel that tensors and so on may form a suitable framework for a coordinate free proof.

• By ”components“ do you mean coordinates relative to some basis or a decomposition of $a$ into components parallel to and perpendicular to the plane of $b$ and $c$? The latter can be without reference to coordinates by using orthogonal projection/rejection. The identity can be derived via these components of $a$. – amd Apr 11 '17 at 5:38
• I was referring to the former. The second approach sounds interesting. I feel the overall nature is similar as they both involve splitting things into components but I'll give that a go when I have a bit more time on my hands! – Harambe Apr 11 '17 at 6:33
• Coordinate-free proofs like this can "easily" be computed using differential forms and particularly the Hodge map. Of course, by "easily", I mean you need a knowledge and understanding of differential forms in the first place. – AloneAndConfused Apr 11 '17 at 7:05
• Could this be what you're after ?math.stackexchange.com/questions/305285/… – Arnaud D. Apr 11 '17 at 7:43

Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ be vector fields on $\mathbb{R}^{3}$ (we could extend to $\mathbb{R}^{n}$ if we wish!), considered as a Riemannian manifold equipped with metric $g$ and induced Hodge map $\star$. Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ have corresponding vector field representations $U,V,W$ on $\mathbb{R}^{n}$ respectively. Then

\begin{align} \mathbf{a}\times(\mathbf{b}\times \mathbf{c}) \,\equiv\, \star(\widetilde{U} \wedge \star(\widetilde{V} \wedge \widetilde{W})) \end{align}

where $\widetilde{X}$ denotes the metric dual of $X$ (i.e. $\widetilde{X}=g(X,-)$) and the equivalence is up to metric dual. Then

\begin{align} \star(\widetilde{U} \wedge \star(\widetilde{V} \wedge \widetilde{W})) &\,=\, \star(\widetilde{U} \wedge i_{W}\star \widetilde{V}) \,=\, \star ( i_{W}\widetilde{U} \wedge \star \widetilde{V} - i_{W} (\widetilde{U}\wedge \star \widetilde{V})) \\ &\,=\, (i_{W}\widetilde{U})\star\star \widetilde{V} - \star i_{W}(\widetilde{U}\wedge\star \widetilde{V}) \\ &\,=\, g(U,W)\widetilde{V} - \star(\widetilde{U}\wedge\star \widetilde{V}) \wedge \widetilde{W} \\ &\,=\, g(U,W)\widetilde{V} - g(U,V)\widetilde{W} \end{align}

where $i_{X}$ denotes the interior derivative with respect to $X$ and we have used the identities:

\begin{align} \star\star \alpha &\,=\, \alpha \\[0.2cm] \star(\widetilde{X}\wedge\star\widetilde{Y}) &\,=\, g(X,Y) \\[0.2cm] \star(\alpha \wedge \widetilde{X}) &\,=\, i_{X}\star \alpha \\[0.2cm] \widetilde{X} \wedge \star\alpha &\,=\, (-1)^{p+1}\star i_{W}\alpha \end{align}

for any $p$-form $\alpha$ and vector fields $X,Y$ (note that the first and second identities are specific to $\mathbb{R}^{3}$). Then note that

$$g(U,W)\equiv \mathbf{a}\cdot\mathbf{c}\quad\text{and}\quad g(U,V)\equiv\mathbf{a}\cdot\mathbf{b}$$

and so then the result follows after taking the metric dual of our expression.

• For now I have no idea what this means, but it's absolutely beautiful. – Harambe Apr 11 '17 at 22:20
• If you are interested, I found the following application-oriented introduction to differential forms extremely useful and it's where the notation can be found: A primer on exterior differential calculus by David A. Burton – AloneAndConfused Apr 12 '17 at 12:28
• it's been over a year, but i've spent the past half year primarily learning differential geometry, and I just came back to this and realised I understood it all :) – Harambe Jun 19 '18 at 11:26

Since $b\times c$ is normal to the plane $b,\,c$ span, $a\times (b\times c)$, which is orthogonal to this vector, is in said plane. The coefficients $B,\,C$ for which the result is $Bb+Cc$ are invariant under rotations, and clearly $B$ must be linear in $a,\,c$ while $C$ is linear in $a,\,b$, so constants $B',\,C'$ exist with $a\times (b\times c) =B' (a\cdot c) b + C' (a\cdot b) c$. Since the left-hand side is antisymmetric, $C'=-B'$. Since $B'$ must be a constant (since both sides are linear in each vector), we can use any vectors we like for which both sides are non-zero to compute $B'$. Example: $a=b=i,\,c=j$ so $a\times (b\times c) = i\times k = -j$ and $(a\cdot c) b - (a\cdot b) c = -j$ as required.

Okay, I really hate the sign issues with the Hodge star, so I am gonna assume that $\star\star=1$, the end result will be good.

The fundamental relationship is that for $k$-vectors/forms, we have $\langle\omega,\eta\rangle\mu=\omega\wedge\star\eta$, where the angle brackets are the inner product on the exterior algebra and $\mu$ is the volume form/multivector.

Let's start with $x$ being an arbitrary vector and taking a look at $$\langle x,a\times(b\times c)\rangle\mu=\langle x,\star(a\wedge\star(b\wedge c))\rangle\mu= \\=x\wedge(a\wedge\star(b\wedge c))=(x\wedge a)\wedge\star(b\wedge c)= \\=\langle x\wedge a,b\wedge c\rangle\mu=\det\left(\begin{matrix}\langle x,b\rangle & \langle x,c\rangle \\ \langle a,b\rangle & \langle a,c\rangle\end{matrix}\right)\mu= \\=(\langle x,b\rangle \langle a,c\rangle-\langle x,c\rangle \langle a,b\rangle)\mu,$$ comparing the LHS with the RHS we can "divide" (ofc not really divide) by $\mu$ since the coefficients need to agree, and because $x$ was arbitrary, and the inner product is nondegenerate, we can "" divide "" by $x$ and we have $$a\times(b\times c)=b\langle a,c\rangle-c\langle a,b\rangle.$$

• I wonder why the inner product of wedge products is a determinant? – Frenzy Li Apr 11 '17 at 7:32
• Also, may cancel be a more appropriate substitute for "divide"? – Frenzy Li Apr 11 '17 at 7:33
• @FrenzyLi Maybe "cancel" would be ok. In the first case, it was because 3-forms in a 3-dim space form a one dimensional space, so equality of 3-forms $\Longleftrightarrow$ equality of the single coefficient, in the second case, I have essentially viewed the inner product as the action of a 1-form (due to Riesz isomorphism and all), and 1-forms agree if their actions on an arbitrary vector agree. As for the determinant, the inner product on the exterior algebra is defined as, if $\alpha=\alpha_1\wedge...\wedge\alpha_k$ and $\beta=\beta_1\wedge... \wedge\beta_k$ then... – Bence Racskó Apr 11 '17 at 7:38
• @FrenzyLi $\langle \alpha,\beta\rangle=\det(\langle \alpha_i,\beta_j\rangle)$. See for example en.wikipedia.org/wiki/…. – Bence Racskó Apr 11 '17 at 7:39
• Nice! And last thing, is $\star\star=1$ generally not true? – Frenzy Li Apr 11 '17 at 7:41

Adapted from my previous proof of $\nabla \times (\vec{A} \times \vec{B})$: \begin{align} \vec a \times (\vec b \times \vec c) & = a_l \hat{e}_l \times (b_i c_j \hat{e}_k \epsilon_{ijk}) \\ & = a_l b_i c_j \epsilon_{ijk} \underbrace{ (\hat{e}_l \times \hat{e}_k)}_{(\hat{e}_l \times \hat{e}_k) = \hat{e}_m \epsilon_{lkm} } \\ & = a_l b_i c_j \hat{e}_m \underbrace{\epsilon_{ijk} \epsilon_{mlk}}_{\text{contracted epsilon identity}} \\ & = a_l b_i c_j \hat{e}_m \underbrace{(\delta_{im} \delta_{jl} - \delta_{il} \delta_{jm})}_{\text{They sift other subscripts}} \\ & = a_j (b_i c_j \hat{e}_i) - a_i (b_i c_j \hat{e}_j) \\ & = (b_i \hat{e}_i) (a_j c_j) - (c_j \hat{e}_j) (a_i b_i) \\ & = \vec b (\vec a\cdot\vec c) - \vec c(\vec a\cdot \vec b) \end{align}

• It uses tensors but in the index notation. Not sure if this is what the OP wants. – Frenzy Li Apr 11 '17 at 4:49
• Yes I've seen a very similar proof before and it's not exactly what I wanted. However, due to what the cross product is, I wouldn't be surprised if all proofs involved index notation or explicit components. (Unless there are "suitable" generalisations!) – Harambe Apr 11 '17 at 5:11
• @Shanye2020 I would actually like to see this proof using more compact notation. I do hope someone could enlighten us. – Frenzy Li Apr 11 '17 at 5:13
• Hmm... An almost identical proof is also seen here. – Frenzy Li Apr 11 '17 at 5:28
• @FrenzyLi Ok, I'm done, posted an answer. – Bence Racskó Apr 11 '17 at 7:28

I just want to add a proof using my favourite method for this kind of thing: Penrose graphical notation. I think it is as coordinate free as you can go.

Every piece of the diagram has a meaning. Tensors are shapes with a lines going upwards or downwards, depending on the type of tensor. For example, vectors have 1 line going upwards, and covectors one line going downwards. Contracion is represented joining the lines.

For example: the first line on the right side, over the word "proof", is the statement $(b\times c)^{a} = b^{b}c^{c}g_{bd}g_{ce}\epsilon^{dea} = b^{b}c^{c}\epsilon_{bcd}g^{da}$ where the indices are proper abstract indices: they do not represent components, but the slots of the tensors.

At the bottom I repeat the properties I use. 1. Antisymmetry of $\epsilon^{abc}$
2. $\epsilon^{abc}\epsilon_{def} = \delta^{abc}_{def}$
3. $\delta^{abc}_{dec} = \delta^{ab}_{de}$
4. $\delta^{ab}_{cd}A_{ab} = 2A_{[cd]} = A_{cd} - A_{dc}$

We can do this by using the definition of the cross product as an bilinear map $$V \times V \to V$$. We also take as a basic property that it is orthogonal to each of its arguments, $$a \cdot (a \times b) = b \cdot (a \times b) = 0 \tag{Orth}$$ We need another property to fix the scaling; the easiest one is that the magnitude of $$a \times b$$ is the area of the parallelogram spanned by $$a$$ and $$b$$. This effectively means that $$(a \times b) \cdot (a \times b) = (a \cdot a)(b \cdot b) - (a \cdot b)^2 \tag{Area}$$ (If this looks opaque, you might prefer to start with perpendicular $$a$$ and $$b$$, then extending with bilinearity.)

(A) and $$v \cdot v = 0 \implies v = 0$$ force that $$a \times a = 0, \tag{Alt}$$ (alterating) and expanding $$(a+b) \times (a+b)$$ and using this implies that $$a \times b = - b \times a : \tag{AntiSym}$$ the cross product must be antisymmetric.

The scalar triple product is $$[a,b,c] = a \cdot (b \times c)$$. It is trilinear since both products are bilinear, and it is also zero when two arguments are equal, by (Orth) and (Alt): $$[a,a,b] = [a,b,a] = [b,a,a] = 0$$ Using linearity on $$[a+b,a+b,c]$$ implies that further $$[a,b,c] = -[b,a,c]$$, and combining this with the inherited antisymmetry in the last two arguments from $$\times$$, we find that $$[a,b,c] = [b,c,a] = [c,a,b] = -[b,a,c] = -[a,c,b] = -[c,b,a] ,$$ and in particular, $$a \cdot (b \times c) = (a \times b) \cdot c :$$ we can swap the position of the dot and the cross.

We can now derive a restricted form of the triple product identity: using linearity on (Area), we have $$0 = ((a + b) \times c) \cdot ((a+b) \times c) + ((a+b) \cdot c)^2 - (a+b) \cdot (a+b))(c \cdot c) \\ = \dotsb = 0 + 0 + (a \times c) \cdot (b \times c) + (a \cdot c)(b \cdot c) - (a \cdot b)(c \cdot c) .$$ Switching the dot and the cross in the first term, we find that $$0 = a \cdot ( c \times (b \times c) + (b \cdot c)c - (c \cdot c)b )$$ But $$a$$ is arbitrary here, so $$c \times (c \times b) = (b \cdot c)c - (c \cdot c)b . \tag{R}$$

We can derive a couple more identities that are true in general, but now we can cheat and use that we are in three dimensions, so we can expand $$a = \lambda b + \mu c + \nu (b \times c)$$. Then using linearity, (R) and linearity again, $$a \times (b \times c) = \lambda (b \times (b \times c)) + \mu ( c \times b \times c ) \\ = \lambda( (c \cdot b)b - (b \cdot b)c ) + \mu ( (c \cdot c)b - (b \cdot c)c ) \\ = ((\lambda b + \mu c) \cdot c)b - ((\lambda b + \mu c) \cdot c)b$$ Finally, we can add the $$\nu (b \times c)$$ part back into both dot products, since it is orthogonal to $$b$$ and $$c$$, which gives $$a \times (b \times c) = (a \cdot c)b - (a \cdot b)c$$ as expected.