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

For what value of $\lambda$ is the volume of the tetrahedron with vertices having position vectors $\hat{i} - 6\hat{j} + 10\hat{k}$,$-\hat{i} - 3\hat{j} + 7\hat{k}$,$5\hat{i} - \hat{j} + \lambda\hat{k}$,and $7\hat{i} - 4\hat{j} + 7\hat{k}$ equal to $11$ cubic units?

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=\hat{i}-2\hat{j}+\hat{k}$,$\vec{c}=\hat{i}+3\hat{j}-2\hat{k}$,and $\vec{d}=2\hat{i}+\hat{j}-\hat{k}$ be four vectors. Let $l=\vec{b} \cdot \vec{c}$ and $m=\vec{b} \cdot \vec{a}$. Find the value of the scalar triple product $[(m\vec{b}+l\vec{a}) \quad \vec{b} \quad \vec{d}]$.

If $\overrightarrow{a} = \alpha \hat{i} + \beta \hat{j} + 3 \hat{k}$,$\overrightarrow{b} = -\beta \hat{i} - \alpha \hat{j} - \hat{k}$,and $\overrightarrow{c} = \hat{i} - 2 \hat{j} - \hat{k}$ such that $\overrightarrow{a} \cdot \overrightarrow{b} = 1$ and $\overrightarrow{b} \cdot \overrightarrow{c} = -3$,then $\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})$ is equal to ............

Let $v = 2i + j - k$ and $w = i + 3k$. If $u$ is any unit vector,then the maximum value of the scalar triple product $[u v w]$ is

Let a vector $\vec{a}$ be coplanar with vectors $\vec{b}=2 \hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to $\vec{d}=3 \hat{i}+2 \hat{j}+6 \hat{k}$,and $|\vec{a}|=\sqrt{10}$. Then a possible value of $[\vec{a} \vec{b} \vec{c}]+[\vec{a} \vec{b} \vec{d}]+[\vec{a} \vec{c} \vec{d}]$ is equal to: