If $\vec{u} = (1, 2)$ and $\vec{v} = (3, 4)$, what is the dot product $\vec{u} \cdot \vec{v}$?
(A) 7
(B) 9
(C) 11
(D) 13
(E) 15

As shown in the figure, in a rectangular parallelepiped $\mathrm { ABCD } - \mathrm { EFGH }$ with $\overline { \mathrm { AB } } = \overline { \mathrm { AD } } = 4$ and $\overline { \mathrm { AE } } = 8$, let P be the point that divides the edge AE in the ratio $1 : 3$, and let Q, R, S be the midpoints of edges $\mathrm { AB }$, $\mathrm { AD }$, and $\mathrm { FG }$, respectively. Let T be the midpoint of segment QR. Find the value of the dot product $\overrightarrow { \mathrm { TP } } \cdot \overrightarrow { \mathrm { QS } }$ of vectors $\overrightarrow { \mathrm { TP } }$ and $\overrightarrow { \mathrm { QS } }$. [3 points]

11. Given that vectors $\overrightarrow { O A } \perp \overrightarrow { O B } , | \overrightarrow { O A } | = 3$, then $\overrightarrow { O A } \bullet \overrightarrow { O B } =$ $\_\_\_\_$.

11. Given vectors $\overrightarrow{OA} \perp \overrightarrow{AB}$ and $|\overrightarrow{OA}| = 3$, then $\overrightarrow{OA} \cdot \overrightarrow{OB} = $ $\_\_\_\_$ .

14. Let vectors $a _ { k } = \left( \cos \frac { k \pi } { 6 } , \sin \frac { k \pi } { 6 } + \cos \frac { k \pi } { 6 } \right) ( k = 0,1,2 , \cdots , 12 )$, then the value of $\sum _ { k = 0 } ^ { 12 } \left( a _ { k } \cdot a _ { k + 1 } \right)$ is $\_\_\_\_$.

4. Given $\vec { a } = ( 0 , - 1 ) , \vec { b } = ( - 1,2 )$, then $( 2 \vec { a } + \vec { b } ) \cdot \vec { a } =$
A. $- 1$
B. $0$
C. $1$
D. $2$

7. Let quadrilateral $A B C D$ be a parallelogram, $| \overrightarrow { A B } | = 6$, $| \overrightarrow { A D } | = 4$. If points $\mathrm { M }$ and $\mathrm { N }$ satisfy $\overrightarrow { B M } = 3 \overrightarrow { M C }$, $\overrightarrow { D N } = 2 \overrightarrow { N C }$, then $\overrightarrow { A M } \cdot \overrightarrow { N M } =$
(A) $20$
(B) $15$
(C) $9$
(D) $6$

13. In isosceles trapezoid $ABCD$, $AB \parallel DC$, $AB = 2$, $BC = 1$, $\angle ABC = 60 ^ { \circ }$. Points $E$ and $F$ are on segments $BC$ and $CD$ respectively, with $\overrightarrow { BE } = \frac { 2 } { 3 } \overrightarrow { BC }$, $\overrightarrow { DF } = \frac { 1 } { 6 } \overrightarrow { DC }$. Then the value of $\overrightarrow { AE } \cdot \overrightarrow { AF }$ is $\_\_\_\_$.

Vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfy $| \boldsymbol { a } | = 1 , \boldsymbol { a } \cdot \boldsymbol { b } = - 1$, then $\boldsymbol { a } \cdot ( 2 \boldsymbol { a } - \boldsymbol { b } ) =$
A. 4
B. 3
C. 2
D. 0

Given vectors $\boldsymbol { a } , \boldsymbol { b }$ satisfying $| \boldsymbol { a } | = 1 , \boldsymbol { a } \cdot \boldsymbol { b } = - 1$, then $\boldsymbol { a } \cdot ( 2 \boldsymbol { a } - \boldsymbol { b } ) =$
A. 4
B. 3
C. 2
D. 0

Given $\overrightarrow { A B } = ( 2,3 ) , \overrightarrow { A C } = ( 3 , t ) , | \overrightarrow { B C } | = 1$, then $\overrightarrow { A B } \cdot \overrightarrow { B C } =$
A． - 3
B． - 2
C． 2
D． 3

Let vectors $\boldsymbol { a }$ and $\boldsymbol { b }$ have an angle whose cosine is $\frac { 1 } { 3 }$, and $| \boldsymbol { a } | = 1$, $| \boldsymbol { b } | = 3$. Then $( 2 \boldsymbol { a } + \boldsymbol { b } ) \cdot \boldsymbol { b } =$ $\_\_\_\_$

Given vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy $|\boldsymbol{a}| = 1, |\boldsymbol{b}| = \sqrt{3}, |\boldsymbol{a} - 2\boldsymbol{b}| = 3$, then $\boldsymbol{a} \cdot \boldsymbol{b} =$
A. $-2$
B. $-1$
C. $1$
D. $2$

Let $\vec{a}, \vec{b}$, and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p\vec{a} + q\vec{b} + r\vec{c}$, where $p$, $q$ and $r$ are scalars, then the value of $\frac{p^2 + 2q^2 + r^2}{q^2}$ is

Column I
(A) In $\mathbb { R } ^ { 2 }$, if the magnitude of the projection vector of the vector $\alpha \hat { i } + \beta \hat { j }$ on $\sqrt { 3 } \hat { i } + \hat { j }$ is $\sqrt { 3 }$ and if $\alpha = 2 + \sqrt { 3 } \beta$, then possible value(s) of $| \alpha |$ is (are)
(B) Let $a$ and $b$ be real numbers such that the function $$f ( x ) = \left\{ \begin{array} { c c } - 3 a x ^ { 2 } - 2 , & x < 1 \\ b x + a ^ { 2 } , & x \geq 1 \end{array} \right.$$ is differentiable for all $x \in \mathbb { R }$. Then possible value(s) of $a$ is (are)
(C) Let $\omega \neq 1$ be a complex cube root of unity. If $\left( 3 - 3 \omega + 2 \omega ^ { 2 } \right) ^ { 4 n + 3 } + \left( 2 + 3 \omega - 3 \omega ^ { 2 } \right) ^ { 4 n + 3 } + \left( - 3 + 2 \omega + 3 \omega ^ { 2 } \right) ^ { 4 n + 3 } = 0$, then possible value(s) of $n$ is (are)
(D) Let the harmonic mean of two positive real numbers $a$ and $b$ be 4. If $q$ is a positive real number such that $a , 5 , q , b$ is an arithmetic progression, then the value(s) of $| q - a |$ is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5

In a triangle $P Q R$, let $\vec { a } = \overrightarrow { Q R } , \vec { b } = \overrightarrow { R P }$ and $\vec { c } = \overrightarrow { P Q }$. If
$$| \vec { a } | = 3 , \quad | \vec { b } | = 4 \quad \text { and } \quad \frac { \vec { a } \cdot ( \vec { c } - \vec { b } ) } { \vec { c } \cdot ( \vec { a } - \vec { b } ) } = \frac { | \vec { a } | } { | \vec { a } | + | \vec { b } | }$$
then the value of $| \vec { a } \times \vec { b } | ^ { 2 }$ is $\_\_\_\_$

If $[ \vec { a } \times \vec { b } \quad \vec { b } \times \vec { c } \quad \vec { c } \times \vec { a } ] = \lambda \left[ \vec { a } \quad \vec { b } \quad \vec { c } \right] ^ { 2 }$ then $\lambda$ is equal to
(1) 0
(2) 1
(3) 2
(4) 3

If $\vec { a } , \vec { b } , \vec { c }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \vec { c } = \overrightarrow { 0 }$, then $| \vec { a } \times \vec { c } |$ is equal to :
(1) $\frac { 1 } { 4 }$
(2) $\frac { 15 } { 16 }$
(3) $\frac { \sqrt { 15 } } { 4 }$
(4) $\frac { \sqrt { 15 } } { 16 }$

If $\vec { a } , \vec { b }$, and $\overrightarrow { \mathrm { c } }$ are unit vectors such that $\vec { a } + 2 \vec { b } + 2 \overrightarrow { \mathbf { c } } = \overrightarrow { 0 }$, then $| \vec { a } \times \overrightarrow { \mathbf { c } } |$ is equal to
(1) $\frac { 1 } { 4 }$
(2) $\frac { \sqrt { 15 } } { 4 }$
(3) $\frac { 15 } { 16 }$
(4) $\frac { \sqrt { 15 } } { 16 }$

Let $\overrightarrow { \mathrm { a } } = \hat { \mathrm { i } } + \alpha \hat { \mathrm { j } } + 3 \hat { \mathrm { k } }$ and $\overrightarrow { \mathrm { b } } = 3 \hat { \mathrm { i } } - \alpha \hat { \mathrm { j } } + \hat { \mathrm { k } }$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec { a }$ and $\vec { b }$ is $8 \sqrt { 3 }$ square units, then $\vec { a } \cdot \vec { b }$ is equal to $\underline{\hspace{1cm}}$.

In a triangle $ABC$, if $| \overrightarrow { BC } | = 3 , | \overrightarrow { CA } | = 5$ and $| \overrightarrow { BA } | = 7$, then the projection of the vector $\overrightarrow { BA }$ on $\overrightarrow { BC }$ is equal to
(1) $\frac { 19 } { 2 }$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 15 } { 2 }$

If $\vec { A } = 2 \hat { \mathrm { i } } + 3 \hat { \mathrm { j } } - \hat { \mathrm { k } }$ m and $\vec { B } = \hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + 2 \hat { \mathrm { k } }$ m. The magnitude of component of vector $\vec { A }$ along vector $\vec { B }$ will be $\_\_\_\_$ m.

If $\vec { a } \cdot \vec { b } = 1 , \vec { b } \cdot \vec { c } = 2$ and $\vec { c } \cdot \vec { a } = 3$, then the value of $[ \vec { a } \times ( \vec { b } \times \vec { c } ) \quad \vec { b } \times ( \vec { c } \times \vec { a } ) \quad \vec { c } \times ( \vec { b } \times \vec { a } ) ]$ is
(1) 0
(2) $- 6 \vec { a } \cdot ( \vec { b } \times \vec { c } )$
(3) $12 \vec { c } \cdot ( \vec { a } \times \vec { b } )$
(4) $- 12 \vec { b } \cdot ( \vec { c } \times \vec { a } )$

Let $\vec { a } = \alpha \hat { i } + \hat { j } - \hat { k }$ and $\vec { b } = 2 \hat { i } + \hat { j } - \alpha \hat { k } , \alpha > 0$. If the projection of $\vec { a } \times \vec { b }$ on the vector $- \hat { i } + 2 \hat { j } - 2 \hat { k }$ is 30, then $\alpha$ is equal to
(1) $\frac { 15 } { 2 }$
(2) 8
(3) $\frac { 13 } { 2 }$
(4) 7

Let $\vec { a } = \alpha \hat { i } + \hat { j } + \beta \hat { k }$ and $\vec { b } = 3 \hat { i } - 5 \hat { j } + 4 \hat { k }$ be two vectors, such that $\vec { a } \times \vec { b } = - \hat { i } + 9 \hat { i } + 12 \widehat { k }$. Then the projection of $\vec { b } - 2 \vec { a }$ on $\vec { b } + \vec { a }$ is equal to
(1) 2
(2) $\frac { 39 } { 5 }$
(3) 9
(4) $\frac { 46 } { 5 }$