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, 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

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