If $$\overrightarrow a = 2\hat i + 2\hat j + 3\hat k,\,\overrightarrow b = - \hat i + 2\hat j + \hat k$$        and $$\overrightarrow c = 3\hat i + \hat j$$   are three vectors such that $$\overrightarrow a + t\overrightarrow b $$   is perpendicular to $$\overrightarrow c ,$$ then what is $$t$$ equal to ?

Solution :
$$\overrightarrow a + t\overrightarrow b = \left( {2 - t} \right)\hat i + \left( {2 + 2t} \right)\hat j + \left( {3 + t} \right)\hat k$$
$$\left( {\overrightarrow a + t\overrightarrow b } \right)$$   and $$\overrightarrow c $$ is perpendicular. Therefore,
$$\eqalign{ & \left( {\overrightarrow a + t\overrightarrow b } \right).\overrightarrow c = 0 \cr & \Rightarrow 3\left( {2 - t} \right) + 2 + 2t = 0 \cr & \Rightarrow 6 - 3t + 2t + 2 = 0 \cr & \Rightarrow t = 8 \cr} $$