If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=2\hat{i}+\lambda\hat{j}+\hat{k}$,$\vec{c}=\hat{i}-\hat{j}+4\hat{k}$ and $\vec{a} \cdot (\vec{b} \times \vec{c}) = 10$,then $\lambda$ is equal to

$|(a \times b) \cdot c| = |a| |b| |c|$,if

Consider the four points $A(1, -2, -1)$, $B(4, 0, -3)$, $C(1, 2, -1)$, and $D(2, -4, -5)$ in space. If $\vec{b} = \vec{AB}$, $\vec{c} = \vec{AC}$, and $\vec{d} = \vec{AD}$, then find the value of $\frac{[\vec{b} \times \vec{c}, \vec{c} \times \vec{d}, \vec{d} \times \vec{b}]}{[\vec{b}+\vec{c}, \vec{c}+\vec{d}, \vec{d}+\vec{b}]}$.

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

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three non-coplanar vectors and $\overrightarrow{p}, \overrightarrow{q}, \overrightarrow{r}$ are defined by the relations $\overrightarrow{p}=\frac{\overrightarrow{b} \times \overrightarrow{c}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}, \quad \overrightarrow{q}=\frac{\overrightarrow{c} \times \overrightarrow{a}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}$ and $\overrightarrow{r}=\frac{\overrightarrow{a} \times \overrightarrow{b}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}$,then $\overrightarrow{a} \cdot \overrightarrow{p}+\overrightarrow{b} \cdot \overrightarrow{q}+\overrightarrow{c} \cdot \overrightarrow{r}$ is equal to

If for vectors $\bar{a}, \bar{b},$ and $\bar{c},$ $[\bar{a} \bar{b} \bar{c}] = 4,$ then $[\bar{a} \times \bar{b}, \bar{b} \times \bar{c}, \bar{c} \times \bar{a}] = \dots$