Given vectors $\overrightarrow { a _ { 1 } } , \overrightarrow { a _ { 2 } } , \overrightarrow { b _ { 1 } } , \overrightarrow { b _ { 2 } } , \ldots , \overrightarrow { b _ { k } } \left( k \in \mathbb{N} ^ { * } \right)$ that are pairwise non-parallel in the plane, satisfying $\left| \overrightarrow { a _ { 1 } } - \overrightarrow { a _ { 2 } } \right| = 1$ and $\left| \overrightarrow { a _ { i } } - \overrightarrow { b _ { j } } \right| \in \{ 1,2 \}$ (where $i = 1,2$ and $j = 1,2 , \ldots , k$), find the maximum value of $k$ as $\_\_\_\_$

Let $P , Q , R$ and $S$ be the points on the plane with position vectors $- 2 \hat { i } - \hat { j } , 4 \hat { i } , 3 \hat { i } + 3 \hat { j }$ and $- 3 \hat { i } + 2 \hat { j }$ respectively. The quadrilateral $P Q R S$ must be a
A) parallelogram, which is neither a rhombus nor a rectangle
B) square
C) rectangle, but not a square
D) rhombus, but not a square

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ} = a\hat{i} + b\hat{j}$ and $\overrightarrow{PS} = a\hat{i} - b\hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\vec{u}$ and $\vec{v}$ be the projection vectors of $\vec{w} = \hat{i} + \hat{j}$ along $\overrightarrow{PQ}$ and $\overrightarrow{PS}$, respectively. If $|\vec{u}| + |\vec{v}| = |\vec{w}|$ and if the area of the parallelogram $PQRS$ is 8, then which of the following statements is/are TRUE?
(A) $a + b = 4$
(B) $a - b = 2$
(C) The length of the diagonal $PR$ of the parallelogram $PQRS$ is 4
(D) $\vec{w}$ is an angle bisector of the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PS}$

Let $a , b , c$ be three distinct real numbers, none equal to one. If the vectors $a \hat { i } + \hat { j } + \widehat { k } , \hat { i } + b \hat { j } + \widehat { k }$ and $\hat { i } + \hat { j } + c \hat { k }$ are coplanar, then $\frac { 1 } { 1 - a } + \frac { 1 } { 1 - b } + \frac { 1 } { 1 - c }$ is equal to
(1) 2
(2) - 1
(3) - 2
(4) 1

9. In coordinate space, three spheres of radius 1 are placed on the $xy$-plane and are mutually tangent to each other. Let their centers be $A, B, C$ respectively. A fourth sphere of radius 1 is placed above these three spheres and is tangent to all three spheres, maintaining stability. Let the center of the fourth sphere be $P$. Which of the following options are correct?
(1) The plane containing points $A, B, C$ is parallel to the $xy$-plane
(2) Triangle $ABC$ is an equilateral triangle
(3) Triangle $PAB$ has one side of length $\sqrt{2}$
(4) The distance from point $P$ to line $AB$ is $\sqrt{3}$
(5) The distance from point $P$ to the $xy$-plane is $1 + \sqrt{3}$