Question

If $\overline{a},\overline{b},\overline{c}$ are three non-coplanar non-zero vectors and $\overline{r}$ is any vector in space, then $(\overline{a}\times \overline{b})\times (\overline{r}\times \overline{c})+(\overline{b}\times \overline{c})\times (\overline{r}\times \overline{a})+(\overline{c}\times \overline{a})\times (\overline{r}\times \overline{b})$ is equal to

• A. $2\left[\overline{a}\overline{b}\overline{c}\right]\overline{r}$
• B. $3\left[\overline{a}\overline{b}\overline{c}\right]\overline{r}$
• C. $\left[\overline{a}\overline{b}\overline{c}\right]\overline{r}$
• D. $4\left[\overline{a}\overline{b}\overline{c}\right]\overline{r}$

Answer 1

The correct option is A $2\left[\overline{a}\overline{b}\overline{c}\right]\overline{r}$

$(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{r}\times \overrightarrow{c})$

$=\left(\right(a\times b).c)r-(a\times b).r)c$

$=\left[abc\right]r-\left[abr\right]c$

Simillarly,

$(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{r}\times \overrightarrow{a})$

$=\left[bca\right]r-\left[bcr\right]a$

and

$(\overrightarrow{c}\times \overrightarrow{a})\times (\overrightarrow{r}\times \overrightarrow{b})$

$=\left[cab\right]r-\left[car\right]b$

When we add all three, we get,

$2\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{r}$