The value of a for which the volume of parallelepiped formed by $\hat{i}+a \hat{j}+\hat{k}, \hat{j}+a \hat{k}$ and $a \hat{i}+\hat{k}$ becomes minimum is

The number of distinct real values of $\lambda$, for which the vectors $-\lambda^2 \hat{i}+\hat{j}+\hat{k}, \hat{i}-\lambda^2 \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\lambda^2 \hat{k}$ are coplanar, is

Let the vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ and $\overline{\mathrm{d}}$ be such that $(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})=\overline{0}$. Let $\mathrm{P}_1$ and $\mathrm{P}_2$ be the planes determined by the pair of vectors $\bar{a}, \bar{b}$ and $\bar{c}, \bar{d}$ respectively, then the angle between $P_1$ and $P_2$ is

Let $\bar{a}, \bar{b}$ and $\overline{\mathrm{c}}$ be three vectors having magnitude 1,1 and 2 respectively. If $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}}=\overline{0}$, then the acute angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ is