If the points having the position vectors $-\hat{i}+4 \hat{j}-4 \hat{k}$,$3 \hat{i}+2 \hat{j}-5 \hat{k}$,$-3 \hat{i}+8 \hat{j}-5 \hat{k}$ and $-3 \hat{i}+2 \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda=$

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - \hat{j}$ and $\vec{c}$ are three vectors such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, then $\vec{c} \cdot (\vec{a} - 2\vec{b})$ is equal to . . . . . . .

If the volume of a parallelepiped,whose coterminous edges are given by the vectors $\overrightarrow{a} = \hat{i} + \hat{j} + n\hat{k}$,$\overrightarrow{b} = 2\hat{i} + 4\hat{j} - n\hat{k}$,and $\overrightarrow{c} = \hat{i} + n\hat{j} + 3\hat{k}$ $(n \geq 0)$,is $158$ cubic units,then which of the following is true?

Let $\vec{c}$ be a vector coplanar with the unit vectors $\vec{a}$ and $\vec{b}$, and let $\vec{d}$ be the unit vector perpendicular to $\vec{a}$, $\vec{b}$, and $\vec{c}$. If $[\vec{a} \vec{b} \vec{d}] \vec{c} - [\vec{a} \vec{b} \vec{c}] \vec{d} = \hat{i} + 2\hat{j} + 2\hat{k}$ and the angle between $\vec{a}$ and $\vec{b}$ is $30^{\circ}$, then $|\vec{c}| =$

If $\bar{a}, \bar{b}, \bar{c}$ are nonzero vectors along the coterminus edges of a parallelepiped with volume $7$ cubic units,then the volume of a parallelepiped with $\bar{a}+\bar{b}, \bar{b}+\bar{c}, \bar{c}+\bar{a}$ as the coterminus edges is

If the vectors $(1 - x)\hat i + \hat j + \hat k$,$\hat i + (1 - y)\hat j + \hat k$,and $\hat i + \hat j + (1 - z)\hat k$ are coplanar,then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is $(x, y, z \neq 0)$.