The proof of Triple Vector Products

Question

THe question is to show the following: $$\vec{a}\times (\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}$$ $$(\vec{a}\times \vec{b})\times\vec{c}=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{b}\cdot\vec{c})\vec{a}$$ In fact, I am able to finish the proof using high school method, which is calculate $\vec{b}\times\vec{c}$ and then cross with $\vec{a}$. However, I would like to use another more mathematical way to prove this triple vector product.

For the first one, $\vec{b}\times\vec{c}$ is a perpendicular vector towards b and c. Then this vector is cross with a. Then, the final results $\vec{a}\times (\vec{b}\times\vec{c})$ is a vector lies on a plane where b and c do also. Hence, it is a linear combination of b and c. $$\vec{a}\times (\vec{b}\times\vec{c})=x\vec{b}+y\vec{c}$$ Take a dot product with $\vec{a} $to both side, L.H.S becomes $0$. $$0=x(\vec{a}\cdot\vec{b})+y(\vec{a}\cdot\vec{c})$$ Then...how an I go further?? I have read a similar proof, but I don't understand the step after I got $0=x(\vec{a}\cdot\vec{b})+y(\vec{a}\cdot\vec{c})$. The reference link is http://www.fen.bilkent.edu.tr/~ercelebi/Ax(BxC).pdf

Thank you.

Answer 1

First, note that we are dealing with polynomial functions here so we can freely impose nonvanishing of certain expressions (as long as there is some $\vec{a},\vec{b},\vec{c}$ that achieves this) and then use a continuity argument to pass to the limiting case. You have already use this when you say you can write the triple product as a linear combination of $\vec{b}$ and $\vec{c}$ (you have thrown away cases where $\vec{b}\parallel\vec{c}$ only to regain them at the limit).

To start with, assume $\vec{a}\cdot\vec{b},\vec{a}\cdot\vec{c}$ are not simultaneously zero. Then the solutions $(x,y)$ to $0=(\vec{a}\cdot\vec{b})x+(\vec{a}\cdot\vec{c})y$ lies on the line perpendicular to $(\vec{a}\cdot\vec{b},\vec{a}\cdot\vec{c})$ and hence you can introduce a constant of proportionality $\lambda$ (a priori could depend on $\vec{a},\vec{b},\vec{c}$) so that $$ (x,y)=\lambda(\vec{a}\cdot\vec{c},-\vec{a}\cdot\vec{b}) $$ and what's more, this constant $\lambda$ can be chosen to be independent of $\vec{a},\vec{b},\vec{c}$. Your reference just assert this and went on to calculate $\lambda$ using specific case. But here is a simpler way to justify part of it that would suffice for the derivation.

We can further restrict our attention to the case $\vec{a}\times(\vec{b}\times\vec{c})\neq\vec{0}$. Then it doesn't hurt to replace $\vec{a}$ by its orthogonal projection along $\vec{b}\times\vec{c}$ to the plane containing $\vec{b}$ and $\vec{c}$ (because it wouldn't change the vector triple product nor $\lambda$). Hence we just need to show that $\lambda$ is the same for $\vec{b}\times(\vec{b}\times\vec{c})$ and $\vec{c}\times(\vec{b}\times\vec{c})$. Taking dot product with $\vec{c}$ in the $\vec{b}\times(\vec{b}\times\vec{c})$ gives $$ [\vec{c},\vec{b},\vec{b}\times\vec{c}]= \lambda(\vec{c}\cdot\vec{b})(\vec{b}\cdot\vec{c})-\lambda(\vec{c}\cdot\vec{c})(\vec{b}\cdot\vec{b}) $$ but the LHS is, by cyclic permutation of the scalar triple product $$ \begin{align*} \vec{c}\cdot(\vec{b}\times(\vec{b}\times\vec{c})) &=(\vec{b}\times\vec{c})\cdot(\vec{c}\times\vec{b})\\ &=-|\vec{b}\times\vec{c}|^2\\ &=-b^2c^2\sin^2\theta\\ &=(bc\cos\theta)^2-b^2c^2\\ &=(\vec{b}\cdot\vec{c})^2-(\vec{b}\cdot\vec{b})(\vec{c}\cdot\vec{c}) \end{align*} $$ So $\lambda=1$ in this case. Similarly, taking scalar product with $\vec{b}$ for $\vec{c}\times(\vec{b}\times\vec{c})$, we get $\lambda=1$ too in this case. So we can take $\lambda=1$ for all $\vec{a},\vec{b},\vec{c}$.

Answer 2

Carrying on from where you left of, \begin{align} y = -\lambda (\vec{a} \cdot \vec{b} ) &\Rightarrow x = \lambda (\vec{a} \cdot \vec{c} ) \\ \vec{a} \times ( \vec{b} \times \vec{c} ) &= \lambda [(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} ] \\ \end{align} Comparing the magnitude of the vectors, gives: \begin{align} ||\vec{a} \times ( \vec{b} \times \vec{c} )||^2 &= ||\lambda [(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} ]||^2 \\ &= \lambda^2 [ (\vec{a} \cdot \vec{c})^2 ||\vec{b} ||^2 + (\vec{a} \cdot \vec{b})^2 ||\vec{c} ||^2 - 2(\vec{a} \cdot \vec{c})(\vec{a} \cdot \vec{b})(\vec{b} \cdot \vec{c} )] \\ LHS &= ||\vec{a} \times ( \vec{b} \times \vec{c} )||^2 \\ &= ||\vec{a} ||^2 ||\vec{b} \times \vec{c} ||^2 - [\vec{a} \ \vec{b} \ \vec{c} ]^2 \\ &= ||\vec{a} ||^2 ( ||\vec{b} ||^2||\vec{c} ||^2 - ( \vec{b} \cdot \vec{c})^2) - [ \ ||\vec{a} ||^2 ||\vec{b} ||^2||\vec{c} ||^2 + 2(\vec{a} \cdot \vec{c})(\vec{a} \cdot \vec{b})(\vec{b} \cdot \vec{c} ) - ||\vec{a}||^2(\vec{b}\cdot\vec{c})^2 - ||\vec{b}||^2(\vec{a}\cdot\vec{c})^2 - ||\vec{c}||^2(\vec{a}\cdot\vec{b})^2] \\ &= (\vec{a} \cdot \vec{c})^2 ||\vec{b} ||^2 + (\vec{a} \cdot \vec{b})^2 ||\vec{c} ||^2 - 2(\vec{a} \cdot \vec{c})(\vec{a} \cdot \vec{b})(\vec{b} \cdot \vec{c} ) \\ &\Rightarrow \lambda^2 = 1 \Rightarrow \lambda = \pm1 \\ \Rightarrow \vec{a} \times ( \vec{b} \times \vec{c} ) &= \pm [(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} ] \end{align} Determining the sign: \begin{align} \vec{a} \times ( \vec{b} \times \vec{c} ) &= \pm [(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} ] \\ \Rightarrow [\vec{a} \times ( \vec{b} \times \vec{c} )] \times \vec{c} &= \pm (\vec{a} \cdot \vec{c}) [\vec{b} \times \vec{c}] \end{align} Using right-hand thumb rule for cross product, it is easy to see that the direction of the LHS and RHS vectors matches with the $+$ sign. \begin{align} \therefore \vec{a} \times ( \vec{b} \times \vec{c} ) &= [(\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} ] \\ (\vec{a} \times \vec{b}) \times \vec{c} &= - [\vec{c} \times ( \vec{a} \times \vec{b} )], \text{ thus solvable.}\\ \end{align}

Note: In the proof of $\lambda = \pm1$, I have used the following properties:

(1) $\vec{p} = \alpha\vec{x} + \beta \vec{y} \Rightarrow ||\vec{p} ||^2 = |\alpha|^2||\vec{x}||^2 + |\beta|^2 ||\vec{y}||^2 + 2\alpha \beta (\vec{x} \cdot \vec{y})$.

(2) Lagrange's Identity: $||\vec{x} \times \vec{y}||^2 + ||\vec{x} \cdot \vec{y}||^2 = ||\vec{x}||^2 ||\vec{y}||^2$

(3) $[\vec{a} \ \vec{b} \ \vec{c} ]^2 = ||\vec{a} \cdot( \vec{b} \times \vec{c}) ||^2 = \ ||\vec{a} ||^2 ||\vec{b} ||^2||\vec{c} ||^2 + 2(\vec{a} \cdot \vec{c})(\vec{a} \cdot \vec{b})(\vec{b} \cdot \vec{c} ) - ||\vec{a}||^2(\vec{b}\cdot\vec{c})^2 - ||\vec{b}||^2(\vec{a}\cdot\vec{c})^2 - ||\vec{c}||^2(\vec{a}\cdot\vec{b})^2$