Value of $\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{i} \cdot(\hat{j} \times \hat{j})+\hat{k} \cdot(\hat{j} \times \hat{i})+\hat{i} \cdot(\hat{k} \times \hat{j})$ is . . . . . . .

If the vectors $\overrightarrow{a}+\lambda \overrightarrow{b}+3 \overrightarrow{c}$,$-2 \overrightarrow{a}+3 \overrightarrow{b}-4 \overrightarrow{c}$ and $\overrightarrow{a}-3 \overrightarrow{b}+5 \overrightarrow{c}$ are coplanar,then the value of $\lambda$ is

If $a, b, c$ are non-coplanar vectors and $\lambda$ is a real number,then the vectors $a + 2b + 3c, \lambda b + 4c$ and $(2\lambda - 1)c$ are non-coplanar for

Let the volume of a parallelepiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k}$,$\overrightarrow{v}=\hat{i}+\hat{j}+3 \hat{k}$ and $\overrightarrow{w}=2 \hat{i}+\hat{j}+\hat{k}$ be $1 \text{ cu. unit}$. If $\theta$ is the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w}$,then $\cos \theta$ can be

If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + aj + 5k$ are coplanar,find the value of $a$.

$(a+b) \cdot(b+c) \times(a+b+c)$ is equal to