Question
$$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$ are three vectors of which every pair is noncollinear. If the vector $$\overrightarrow a + \overrightarrow b $$ and $$\overrightarrow b + \overrightarrow c $$ are collinear with $$\overrightarrow c $$ and $$\overrightarrow a $$ respectively then $$\overrightarrow a + \overrightarrow b + \overrightarrow c $$ is :
A.
a unit vector
B.
the null vector
C.
equally inclined to $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$
D.
none of these
Answer :
the null vector
Solution :
Here, $$\overrightarrow a + \overrightarrow b = t\overrightarrow c ,\,\,\overrightarrow b + \overrightarrow c = s\overrightarrow a .$$ Subtracting, $$\overrightarrow a - \overrightarrow c = t\overrightarrow c - s\overrightarrow a $$
or $$\left( {1 + s} \right)\overrightarrow a = \left( {1 + t} \right)\overrightarrow c $$
But $$\overrightarrow a ,\,\overrightarrow c $$ are noncollinear.
$$\therefore \,1 + s = 0,\,\,1 + t = 0.$$ Hence, $$\overrightarrow a + \overrightarrow b = - \overrightarrow c $$