If $ABCDEF$ is a regular hexagon,then $\overrightarrow{AD} + \overrightarrow{EB} + \overrightarrow{FC} = .....$

The vector that is parallel to the vector $2 \hat{i} - 2 \hat{j} - 4 \hat{k}$ and coplanar with the vectors $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$ is

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors in the $xyz$-space such that $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \neq 0$. If $A, B, C$ are points with position vectors $\vec{a}, \vec{b}, \vec{c}$ respectively,then the number of possible positions of the centroid of $\triangle ABC$ is

In the given figure,if a vector $x$ satisfies the equation $x - w = v$,then $x = ?$

The minimum value of $(x_1 - x_2)^2 + (\sqrt{2 - x_1^2} - \frac{9}{x_2})^2$ where $x_1 \in (0, \sqrt{2})$ and $x_2 \in R^+$.

If $ABCDEF$ is a regular hexagon,and $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} = k \vec{AD}$,then $k = \dots$