If $\bar{u}=2\hat{i}+3\hat{j}+4\hat{k}$, $\bar{v}=\hat{i}+4\hat{j}+3\hat{k}$ and $\bar{w}=\hat{i}+7\hat{j}+\lambda\hat{k}$ represent the sides of a triangle, find the value of $\lambda$.