Question

For three vectors $\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$ which of the following expressions is no equal to any one of the remaining three option?

• A. $\overrightarrow{u}\cdot (\overrightarrow{v}\times \overrightarrow{w})$
• B. $(\overrightarrow{v}\times \overrightarrow{w})\cdot \overrightarrow{u}$
• C. $\overrightarrow{v}\cdot (\overrightarrow{u}\times \overrightarrow{w})$
• D. $(\overrightarrow{u}\times \overrightarrow{v})\cdot \overrightarrow{w}$

Answer 1

The correct option is C $\overrightarrow{v}\cdot (\overrightarrow{u}\times \overrightarrow{w})$

First, dot product is commutative and thus $\overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{b}\cdot \overrightarrow{a}$
Also, scalar triple product is cyclic and so $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{a}\right]=\left[\overrightarrow{c}\overrightarrow{a}\overrightarrow{b}\right]$
Using these, we have option c as the answer.