The points $A(4, 5, 1)$,$B(0, -1, -1)$,$C(3, 9, 4)$,and $D(-4, 4, 4)$ are:

If $a=x \hat{i}+y \hat{j}+z \hat{k}$, then $(a \times \hat{i}) \cdot(\hat{i}+\hat{j})+(a \times \hat{j}) \cdot(\hat{j}+\hat{k})+(a \times \hat{k}) \cdot(\hat{k}+\hat{i})=$

If vectors $\vec{A} = 2i + 3j + 4k$,$\vec{B} = i + j + 5k$,and $\vec{C}$ form a left-handed system,then $\vec{C}$ is

If $[\bar{a} \times \bar{b} \quad \bar{b} \times \bar{c} \quad \bar{c} \times \bar{a}] = \lambda [\bar{a} \quad \bar{b} \quad \bar{c}]^2$,then $\lambda$ is equal to

Let $\vec{b} = -\hat{i} + 4\hat{j} + 6\hat{k}$ and $\vec{c} = 2\hat{i} - 7\hat{j} - 10\hat{k}$. If $\vec{a}$ is a unit vector and the scalar triple product $[\vec{a} \ \vec{b} \ \vec{c}]$ has the greatest value,then $\vec{a}$ is equal to:

If the volume of the parallelopiped with $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}$ and $\vec{c} \times \vec{a}$ as coterminous edges is $9 \text{ cu. units}$, then the volume of the parallelopiped with $(\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})$ and $(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})$ as coterminous edges is