Question
Let $$\overrightarrow \lambda = \overrightarrow a \times \left( {\overrightarrow b + \overrightarrow c } \right),\,\overrightarrow \mu = \overrightarrow b \times \left( {\overrightarrow c + \overrightarrow a } \right)$$ and $$\overrightarrow \nu = \overrightarrow c \times \left( {\overrightarrow a + \overrightarrow b } \right).$$ Then :
A.
$$\overrightarrow \lambda + \overrightarrow \mu = \overrightarrow \nu $$
B.
$$\overrightarrow \lambda ,\,\overrightarrow \mu ,\,\overrightarrow \nu $$ are coplanar
C.
$$\overrightarrow \lambda + \overrightarrow \nu = 2\overrightarrow \mu $$
D.
none of these
Answer :
$$\overrightarrow \lambda ,\,\overrightarrow \mu ,\,\overrightarrow \nu $$ are coplanar
Solution :
Adding and simplifying, $$\overrightarrow \lambda + \overrightarrow \mu + \overrightarrow \nu = 0$$
$$\therefore $$ there is a linear relation between $$\overrightarrow \lambda ,\,\overrightarrow \mu ,\,\overrightarrow \nu $$ and, therefore, they are coplanar.