If $$\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$$ and $$\overline{\mathrm{c}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}$$ are such that $$\bar{a}+\lambda \bar{b}$$ is perpendicular to $$\bar{c}$$, then the value of $$\lambda$$ is

If $$\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$$ and $$\overline{\mathrm{c}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\beta \hat{\mathrm{k}}$$ are linearly dependent vectors and $$|\bar{c}|=\sqrt{3}$$, then the values of $$\alpha$$ and $$\beta$$ are respectively.

If the volume of the parallelopiped is $$158 \mathrm{~cu}$$. units whose coterminus edges are given by the vectors $$\bar{a}=(\hat{i}+\hat{j}+n \hat{k}), \bar{b}=2 \hat{i}+4 \hat{j}-n \hat{k}$$ and $$\bar{c}=\hat{i}+n \hat{j}+3 \hat{k}$$, where $$n \geq 0$$, then the value of $$n$$ is

If $$\bar{a}, \bar{b}, \bar{c}$$ are three vectors such that $$\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}}+\overline{\mathrm{c}})+\overline{\mathrm{b}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})+\overline{\mathrm{c}} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})=0 \quad$$ and $$\quad|\overline{\mathrm{a}}|=1$$, $$|\bar{b}|=8$$ and $$|\bar{c}|=4$$, then $$|\bar{a}+\bar{b}+\bar{c}|$$ has the value _________.