# If vectros $\overrightarrow a = 2\hat i + 2\hat j +3\hat k , \overrightarrow b = -\hat i + 2\hat j +\hat k \: and \: \overrightarrow c = 3\hat i + \hat j$ are such that $( \overrightarrow a +\lambda \overrightarrow b)$ is perpendicular to $( \overrightarrow c )$, then find the value of $( \lambda )$

$\begin{array}{1 1}\text{a. 8} \\ \text{b. 6} \\ \text{c. 9} \\ \text{d. 5} \end{array}$

Let $\overrightarrow a = 2\hat i + 2\hat j +3\hat k , \overrightarrow b = -\hat i + 2\hat j +\hat k \: and \: \overrightarrow c = 3\hat i + \hat j$
$\lambda \overrightarrow b = - \lambda \hat i + 2 \lambda \hat j + \lambda \hat k$
$\overrightarrow a + \lambda \overrightarrow b = (2 - \lambda) \hat i + (2 + 2 \lambda) \hat j + (3 + \lambda) \hat k$
$(\overrightarrow a + \lambda \overrightarrow b) . \overrightarrow c = 0$
$(2 - \lambda) 3 + (2 + 2\lambda) = 0$
$6 - 3\lambda + 2 + 2\lambda = 0$
$8 - \lambda = 0$
$8 = \lambda$