Let $\bar{a}=\hat{i}+\hat{j}, \bar{b}=2 \hat{i}-\hat{k}, \bar{c}=3 \hat{i}-\hat{j}+\hat{k}$, then vector $\overline{\mathrm{p}}$ satisfying $\overline{\mathrm{p}} \cdot \overline{\mathrm{a}}=0$ and $\overline{\mathrm{p}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}} \times \overline{\mathrm{b}}$ is

If $\bar{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \bar{b}=\hat{i}-2 \hat{j}-2 \hat{k}, \bar{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$ and if $\overline{\mathrm{d}}$ is vector perpendicular to both $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}, \overline{\mathrm{a}} \cdot \overline{\mathrm{d}}=18$, then $|\overline{\mathrm{a}} \times \overline{\mathrm{d}}|^2=$

Let $\bar{a}, \bar{b}, \bar{c}$ be three vectors such that $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}=\overline{0},|\overline{\mathrm{a}}|=3,|\overline{\mathrm{~b}}|=4,|\overline{\mathrm{c}}|=5$, then $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}=$

If $\bar{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\bar{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$, then the value of $(2 \overline{\mathrm{a}}-\overline{\mathrm{b}}) \cdot((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{a}}+2 \overline{\mathrm{~b}}))=$