In the analytic plane, the sides of a regular pentagon are named as vectors $\vec { a } , \vec { b } , \vec { c } , \vec { d }$ and $\vec { e }$ as shown in the figure.
Accordingly, what is the probability that the dot product of two vectors randomly selected from these five vectors is positive?
A) $\frac { 1 } { 2 }$ B) $\frac { 1 } { 5 }$ C) $\frac { 2 } { 5 }$ D) $\frac { 1 } { 10 }$ E) $\frac { 3 } { 10 }$

In the Cartesian coordinate plane, vectors $\overrightarrow { \mathrm { u } _ { 1 } } = ( 3,4 )$ and $\overrightarrow { \mathrm { u } _ { 2 } } = ( 8 , - 6 )$ are given. For a vector $\vec { V }$ taken in this plane, the perpendicular projection vector onto the $\overrightarrow { u _ { 1 } }$ vector is 3 units, and the perpendicular projection vector onto the $\overrightarrow { u _ { 2 } }$ vector is 1 unit in length.
Accordingly, what is the length of the $\vec { v }$ vector in units?
A) $\sqrt { 5 }$ B) $\sqrt { 10 }$ C) $5 \sqrt { 5 }$ D) 5 E) 10

Three cubes, each with edge length 1 unit, are glued together such that at least one face of each cube completely overlaps with a face of another cube.
Accordingly, which of the following cannot be the distance between two selected vertices of the solid obtained in this way, in units?
A) $\sqrt { 7 }$ B) $\sqrt { 8 }$ C) $\sqrt { 9 }$ D) $\sqrt { 10 }$ E) $\sqrt { 11 }$

In the rectangular coordinate plane, a point $P ( a , b )$ is rotated counterclockwise by $90 ^ { \circ }$ around the origin, and then the resulting point is translated 3 units in the positive direction along the x-axis and 1 unit in the positive direction along the y-axis, yielding the point $P ( a , b )$ again. Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 0
B) 1
C) 2
D) 3
E) 4