If $ABCDEF$ is a regular hexagon,the length of whose side is $a$,then $\overrightarrow{AB} \cdot \overrightarrow{AF} + \frac{1}{2} \overrightarrow{BC}^2 = $

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \overrightarrow{a} + 2 \overrightarrow{b}$ and $3 \overrightarrow{a} - \lambda \overrightarrow{b}$ are perpendicular to each other,then the number of values of $\lambda$ in $[-1, 3]$ is:

The vector $c$ directed along the internal bisector of the angle between the vectors $a = 7i - 4j - 4k$ and $b = -2i - j + 2k$ with $|c| = 5\sqrt{6}$ is

The points $A(5, -1, 1)$,$B(7, -4, 7)$,$C(1, -6, 10)$,and $D(-1, -3, 4)$ are vertices of a:

Let $v_1, v_2, v_3, v_4$ be unit vectors in the $XY$-plane,one each in the interior of the four quadrants. Which of the following statements is necessarily true?

If $a$ makes an acute angle with $b$,$r \cdot a = 0$ and $r \times b = c \times b$,then $r=$