If the coordinates of points $A, B, C,$ and $D$ are $(1, 2, 3), (4, 5, 7), (-4, 3, -6),$ and $(2, 9, 2)$ respectively,then the angle between $AB$ and $CD$ is:

If $a = 2i + j - 8k$ and $b = i + 3j - 4k,$ then the magnitude of $a + b = $

If the position vectors of $A$ and $B$ are respectively $(1, 1, 0)$ and $(0, 1, 1)$,then $\overrightarrow{AB} =$

Given vectors $\vec{a} = 2 \hat{i} - \hat{j} + 2 \hat{k}$ and $\vec{b} = -\hat{i} + \hat{j} - \hat{k}$. The vector in the direction of $\vec{a} + \vec{b}$ with magnitude $\sqrt{2}$ is . . . . . . .

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

Let $\bar{a}$ and $\bar{b}$ be the position vectors of points $A$ and $B$ respectively. $C$ and $D$ are points on the line $AB$ such that $\overline{AC} = 3 \overline{AB}$ and $\overline{BD} = 2 \overline{BA}$. Find the vector $\overline{CD}$.