Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}, \overline{\mathrm{d}}$ are vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{c}} \times \overline{\mathrm{d}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}$ and if $\left|\begin{array}{ll}\overline{\mathrm{a}} \cdot \overline{\mathrm{c}} & \overline{\mathrm{b}} \cdot \overline{\mathrm{c}} \\ \overline{\mathrm{a}} \cdot \overline{\mathrm{d}} & \overline{\mathrm{b}} \cdot \overline{\mathrm{d}}\end{array}\right|=0$, then $\lambda=$

Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}$ and $\bar{c}=\hat{j}-\hat{k}$ be three vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=1$. If the length of projection vector of the vector $\overline{\mathrm{b}}$ on the vector $\overline{\mathrm{a}} \times \overline{\mathrm{c}}$ is $l$, then the value of $3 l^2$ is

The area of the rectangle having vertices $\mathrm{P}, \quad \mathrm{Q}, \quad \mathrm{R}, \quad \mathrm{S}$ with position vectors $-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}},-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ respectively is

If $\bar{a}, \bar{b}, \bar{c}$ are three vectors such that $|\overrightarrow{\mathrm{a}}|=\sqrt{31}, 4|\overrightarrow{\mathrm{~b}}|=|\overrightarrow{\mathrm{c}}|=2$ and $2(\overline{\mathrm{a}} \times \overline{\mathrm{b}})=3(\overline{\mathrm{c}} \times \overline{\mathrm{a}})$ and if the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $\frac{2 \pi}{3}$ then $\left|\frac{\bar{a} \times \bar{c}}{\bar{a} \cdot \bar{b}}\right|^2=$