If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = 2\hat{i} - 4\hat{k}$,and $\vec{c} = \hat{i} + \lambda \hat{j} + 3\hat{k}$ are coplanar,then the value of $\lambda$ is:

If $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are non-zero coplanar vectors,then $[2 \overrightarrow{a}-\overrightarrow{b} \quad 3 \overrightarrow{b}-\overrightarrow{c} \quad 4 \overrightarrow{c}-\overrightarrow{a}]$ is

The sum of the distinct real values of $\mu$,for which the vectors $\mu \hat{i} + \hat{j} + \hat{k}$,$\hat{i} + \mu \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + \mu \hat{k}$ are coplanar,is

Let $p, q, r$ be three non-coplanar vectors and $b = p \times q$. If $a, b, c$ denote the coterminous edges of a parallelepiped, then its height with the base having $a$ and $c$ is

If $\bar{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k}, \bar{b}=2 \hat{\imath}-\hat{\jmath}+7 \hat{k}$ and $\bar{c}=7 \hat{\imath}-\hat{\jmath}+23 \hat{k}$ are three vectors,then which of the following statements is true?

If $a = i - j + k$,$b = i + 2j - k$ and $c = 3i + pj + 5k$ are coplanar,then the value of $p$ is: