If $\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j}$ are such that the vector $(\vec{a} + \lambda \vec{b})$ is perpendicular to vector $\vec{c}$, then find the value of $\lambda$.

$8$

The correct answer is Option (1) → $8$ ##

If $\vec{a} + \lambda\vec{b}$ is perpendicular to $\vec{c}$, then

$(\vec{a} + \lambda\vec{b}) \cdot \vec{c} = 0$

$\Rightarrow \{(2-\lambda)\hat{i} + (2+2\lambda)\hat{j} + (3+\lambda)\hat{k}\} \cdot (3\hat{i} + \hat{j})= 0$

$\Rightarrow (2-\lambda) \cdot 3 + (2+2\lambda) \cdot 1 + (3+\lambda) \cdot 0 = 0$

$\Rightarrow 6 - 3\lambda + 2 + 2\lambda = 0$

$\Rightarrow -\lambda + 8 = 0$

$\Rightarrow \lambda = 8$