Question
If $$\overrightarrow r = 3\overrightarrow i + 2\overrightarrow j - 5\overrightarrow k ,\,\overrightarrow a = 2\overrightarrow i - \overrightarrow j + \overrightarrow k ,\,\overrightarrow b = \overrightarrow i + 3\overrightarrow j - 2\overrightarrow k $$ and $$\overrightarrow c = - 2\overrightarrow i + \overrightarrow j - 3\overrightarrow k $$ such that $$\overrightarrow r = \lambda \overrightarrow a + \mu \overrightarrow b + \nu \overrightarrow c $$ then :
A.
$$\mu ,\,\frac{\lambda }{2},\,\nu $$ are in AP
B.
$$\lambda ,\,\mu ,\,\nu $$ are in AP
C.
$$\lambda ,\,\mu ,\,\nu $$ are in HP
D.
$$\mu ,\,\lambda ,\,\nu $$ are in GP
Answer :
$$\mu ,\,\frac{\lambda }{2},\,\nu $$ are in AP
Solution :
Here,
$$\eqalign{
& 3\overrightarrow i + 2\overrightarrow j - 5\overrightarrow k = \lambda \left( {2\overrightarrow i - \overrightarrow j + \overrightarrow k } \right) + \mu \left( {\overrightarrow i + 3\overrightarrow j - 2\overrightarrow k } \right) + \nu \left( { - 2\overrightarrow i + \overrightarrow j - 3\overrightarrow k } \right) \cr
& \Rightarrow 3 = 2\lambda + \mu - 2\nu ,\,\,2 = - \lambda + 3\mu + \nu ,\,\, - 5 = \lambda - 2\mu - 3\nu \cr} $$
Solving these, $$\mu = 1,\,\nu = 2,\,\lambda = 3.$$ Hence, $$\mu ,\,\frac{\lambda }{2},\,\nu $$ are in AP.