Find the value of $\lambda$ such that the vectors $\mathbf{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\mathbf{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal.

$\frac{-5}{2}$

The correct answer is Option (4) → $\frac{-5}{2}$ ##

Since, two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal i.e., $\mathbf{a} \cdot \mathbf{b} = 0$.

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

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

$\Rightarrow \lambda = \frac{-5}{2}$