Prove that if $\:$$\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{r}$ are any four vectors and if $[\vec{x} \: \vec{y} \: \vec{z}]$ is Scalar Triple product, Then
$$[\vec{a} \: \vec{b} \: \vec{r}]\vec{c}+[\vec{b} \: \vec{c} \: \vec{r}]\vec{a}+[\vec{c} \: \vec{a} \: \vec{r}]\vec{b}=[\vec{a} \: \vec{b} \: \vec{c}]\vec{r}$$
My Try: Let $$F(\vec{r})=[\vec{a} \: \vec{b} \: \vec{r}]\vec{c}+[\vec{b} \: \vec{c} \: \vec{r}]\vec{a}+[\vec{c} \: \vec{a} \: \vec{r}]\vec{b}$$
where $F(.)$ is a function.
Put $\vec{r}=\vec{a}$ Then
$$F(\vec{a})=[\vec{a} \: \vec{b} \: \vec{c}]\vec{a}$$ Put $\vec{r}=\vec{b} $ Then
$$F(\vec{b})=[\vec{a} \: \vec{b} \: \vec{c}]\vec{b}$$ Similarly
put $\vec{r}=\vec{c}$ Then
$$F(\vec{c})=[\vec{a} \: \vec{b} \: \vec{c}]\vec{c}$$
From the three equations above, we can observe that $F(.)$ is a linear function with slope $[\vec{a} \: \vec{b} \: \vec{c}]$ Hence
$$F(\vec{r})=[\vec{a} \: \vec{b} \: \vec{c}]\vec{r}$$
I feel this approach is quite informal..can any one give me better way to prove this.