Practicing Success
For three vectors $\vec u, \vec v, \vec w$ which of the following expressions is not equal to any of the remaining three? |
$\vec u.(\vec v×\vec w)$ $(\vec v×\vec w).\vec u$ $\vec v.(\vec u×\vec w)$ $(\vec u×\vec v).\vec w$ |
$\vec v.(\vec u×\vec w)$ |
We have, $\vec u.(\vec v×\vec w)=(\vec v×\vec w).\vec u$ [∵ Dot product is commutative] So, expression in options (1) and (2) are equal. Since the positions of dot and cross can be interchanged in scalar triple product. Therefore, $(\vec u×\vec v).\vec w=\vec u.(\vec v×\vec w)$ So, expression in options (1) and (4) are equal. But, $\vec v.(\vec u×\vec w) = (\vec u×\vec w). \vec v = \vec u. (\vec w×\vec v) = - \vec u. (\vec v×\vec w)$ Hence, expression in (3) is not equal to the remaining three. |