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]

12. In equilateral triangle $ABC$, $D$ is a point on side $BC$. If $AB = 3, BD = 1$, then $\overrightarrow{AB} \cdot \overrightarrow{AD} =$ $\_\_\_\_$

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. $2$

15. (a) Find 3 -dimensional vectors $\overrightarrow { v _ { 1 } } , \overrightarrow { v _ { 2 } } , \overrightarrow { v _ { 3 } }$ satisfying
$$\overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 1 } } = 4 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 2 } } = - 2 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 3 } } = 6 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 2 } } = 2 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 3 } } = - 5 , \overrightarrow { v _ { 3 } } \cdot \overrightarrow { v _ { 3 } } = 29 .$$
(b) Let $\vec { A } ( t ) = f _ { 1 } ( t ) \hat { i } + f _ { 2 } ( t ) \hat { j }$ and $B ( t ) = g _ { 1 } ( t ) \hat { i } + g _ { 2 } ( t ) \hat { j } , t \in [ 0,1 ]$, where $f _ { 1 \text {, } } \mathrm { f } _ { 2 } , \mathrm {~g} _ { 1 } , \mathrm {~g} _ { 2 }$ are continuous functions. If $\vec { A } ( \mathrm { t } )$ and $\vec { B } ( \mathrm { t } )$ are non-zero vectors for all t and $\vec { A } ( 0 ) = 2 \hat { \imath } + 3 \hat { \jmath } , \vec { A } ( 1 ) = 6 \hat { \imath } + 2 \hat { \jmath } , \vec { B } ( 0 ) = 3 \hat { \imath } + 2 \hat { \jmath }$ and $\vec { B } ( 1 ) = 2 \hat { \imath } + 6 \hat { j }$. The show that $\vec { A } ( t )$ and $\vec { B } ( t )$ are parallel for some $t$.

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

Let $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$. Let $\vec{c}$ be a vector such that $|\vec{c} - \vec{a}| = 3$, $|(\vec{a} \times \vec{b}) \times \vec{c}| = 3$ and the angle between $\vec{c}$ and $\vec{a} \times \vec{b}$ is $30^\circ$. Then $\vec{a} \cdot \vec{c}$ is equal to:
(1) $\frac{1}{8}$
(2) $25$
(3) $2$
(4) $5$

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.