First slide

Vector triple product

Question

Let $\vec{a}, \vec{b} and \vec{c}$ be three non-coplanar vectors and $\vec{r}$ be any arbitrary vector. Then $(\vec{a} \times \vec{b}) \times (\vec{r} \times \vec{c}) + (\vec{b} \times \vec{c}) \times (\vec{r} \times \vec{a}) + (\vec{c} \times \vec{a}) \times (\vec{r} \times \vec{b})$ i, always equal to

Moderate

A

$[\vec{a} \vec{b} \vec{c}] \vec{r}$
B

$2 [\vec{a} \vec{b} \vec{c}] \vec{r}$
C

$3 [\vec{a} \vec{b} \vec{c}] \vec{r}$
D

none of these

Solution

$\begin{array}{l} (\vec{a} \times \vec{b}) \times (\vec{r} \times \vec{c}) \\ = ((\vec{a} \times \vec{b}) \cdot \vec{c}) \vec{r} - ((\vec{a} \times \vec{b}) \cdot \vec{r}) \cdot \vec{c} \\ = [\vec{a} \vec{b} \vec{c}] \vec{r} - [\vec{a} \vec{b} \vec{r}] \vec{c} \\ Similarly, (\vec{b} \times \vec{c}) \times (\vec{r} \times \vec{a}) \\ = [\vec{b} \vec{c} \vec{a}] \vec{r} - [\vec{b} \vec{c} \vec{r}] \vec{a} \\ and (\vec{c} \times \vec{a}) \times (\vec{r} \times \vec{b}) = [\vec{c} \vec{a} \vec{b}] \vec{r} - [\vec{c} \vec{a} \vec{r}] \vec{b} \\ \Rightarrow (\vec{a} \times \vec{b}) \times (\vec{r} \times \vec{c}) + (\vec{b} \times \vec{c}) \times (\vec{r} \times \vec{a}) + (\vec{c} \times \vec{a}) \times (\vec{r} \times \vec{b}) \\ = 3 [\vec{a} \vec{b} \vec{c}] \vec{r} - ([\vec{b} \vec{c} \vec{r}] \vec{a} + [\vec{c} \vec{a} \vec{r}] \vec{b} + [\vec{a} \vec{b} \vec{r}] \vec{c}) \\ = 3 [\vec{a} \vec{b} \vec{c}] \vec{r} - [\vec{a} \vec{b} \vec{c}] \vec{r} = 2 [\vec{a} \vec{b} \vec{c}] \vec{r} \end{array}$

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