In coordinate space, there are two points $\mathrm { A } ( 3,1,1 ) , \mathrm { B } ( 1 , - 3 , - 1 )$. For a point P on the plane $x - y + z = 0$, what is the minimum value of $| \overrightarrow { \mathrm { PA } } + \overrightarrow { \mathrm { PB } } |$? [4 points]
(1) $\frac { 4 \sqrt { 3 } } { 3 }$
(2) $\frac { 5 \sqrt { 3 } } { 3 }$
(3) $2 \sqrt { 3 }$
(4) $\frac { 7 \sqrt { 3 } } { 3 }$
(5) $\frac { 8 \sqrt { 3 } } { 3 }$

In coordinate space, let $l$ be the line of intersection of the plane $x = 3$ and the plane $z = 1$. When point P moves on line $l$, what is the minimum value of the length of segment OP? (Here, O is the origin.) [3 points]
(1) $2 \sqrt { 2 }$
(2) $\sqrt { 10 }$
(3) $2 \sqrt { 3 }$
(4) $\sqrt { 14 }$
(5) $3 \sqrt { 2 }$

In coordinate space, a line $l : \frac { x } { 2 } = 6 - y = z - 6$ and plane $\alpha$ meet perpendicularly at point $\mathrm { P } ( 2,5,7 )$. For a point $\mathrm { A } ( a , b , c )$ on line $l$ and a point Q on plane $\alpha$, when $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } } = 6$, what is the value of $a + b + c$? (Here, $a > 0$) [4 points]
(1) 15
(2) 16
(3) 17
(4) 18
(5) 19

The point $T ( 7 | 10 | 0 )$ lies on the edge $\left[ \mathrm { A } _ { 3 } \mathrm {~A} _ { 4 } \right]$. Investigate computationally whether there are points on the edge $\left[ \mathrm { B } _ { 3 } \mathrm {~B} _ { 4 } \right]$ for which the following holds: The line segments connecting the point to the points $B _ { 1 }$ and $T$ are perpendicular to each other. If applicable, give the coordinates of these points.

Water fountains emerge at four points on the surface of the marble sphere. One of these exit points is described in the model by the point $L _ { 0 } ( 1 | 1 | 6 )$. The corresponding fountain is modeled by points $L _ { t } \left( t + 1 | t + 1 | 6,2 - 5 \cdot ( t - 0,2 ) ^ { 2 } \right)$ with suitable values $t \in \mathbb { R } _ { 0 } ^ { + }$.
Investigate whether the highest point of the water fountain is higher than the highest point of the fountain.

Let $P$ and $Q$ be the points on the line $\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$ which are at a distance of 6 units from the point $R(1,2,3)$. If the centroid of the triangle $PQR$ is $(\alpha, \beta, \gamma)$, then $\alpha^2 + \beta^2 + \gamma^2$ is:
(1) 26
(2) 36
(3) 18
(4) 24

11. In coordinate space, the point on line $L$ closest to point $Q$ is called the projection of $Q$ onto $L$. Given that $L$ is a line on the plane $2x - y = 2$ passing through the point $(2, 2, 2)$. Which of the following points could be the projection of the origin $O$ onto $L$?
(1) $(2, 2, 2)$
(2) $(2, 0, 2)$
(3) $\left(\frac{4}{5}, -\frac{2}{5}, 0\right)$
(4) $\left(\frac{4}{5}, -\frac{2}{5}, -2\right)$
(5) $\left(\frac{8}{9}, -\frac{2}{9}, -\frac{2}{9}\right)$