As shown in the figure, on a plane $\alpha$ there is an equilateral triangle ABC with side length 3, and a sphere $S$ with radius 2 is tangent to the plane $\alpha$ at point A. For a point D on the sphere $S$ such that the segment AD passes through the center O of the sphere $S$, find the value of $| \overrightarrow { \mathrm { AB } } + \overrightarrow { \mathrm { DC } } | ^ { 2 }$. [4 points]

In coordinate space, the distance between point $\mathrm { P } ( 0,3,0 )$ and point $\mathrm { A } ( - 1,1 , a )$ is 2 times the distance between point P and point $\mathrm { B } ( 1,2 , - 1 )$. What is the value of the positive number $a$? [2 points]
(1) $\sqrt { 7 }$
(2) $\sqrt { 6 }$
(3) $\sqrt { 5 }$
(4) 2
(5) $\sqrt { 3 }$

In triangle ABC,
$$\overline { \mathrm { AB } } = 2 , \quad \angle \mathrm {~B} = 90 ^ { \circ } , \quad \angle \mathrm { C } = 30 ^ { \circ }$$
When point P satisfies $\overrightarrow { \mathrm { PB } } + \overrightarrow { \mathrm { PC } } = \overrightarrow { 0 }$, what is the value of $| \overrightarrow { \mathrm { PA } } | ^ { 2 }$? [3 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

For two vectors $\vec { a } = ( 1,3 ) , \vec { b } = ( 5 , - 6 )$, what is the sum of all components of the vector $\vec { a } - \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two vectors $\vec { a } = ( 1 , - 2 ) , \vec { b } = ( - 1,4 )$, what is the sum of all components of the vector $\vec { a } + 2 \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two vectors $\vec { a } = ( 3,1 ) , \vec { b } = ( - 2,4 )$, what is the sum of all components of the vector $\vec { a } + \frac { 1 } { 2 } \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Four distinct points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ on a circle satisfy the following conditions. What is the value of $| \overrightarrow { \mathrm { AD } } | ^ { 2 }$? [4 points] (가) $| \overrightarrow { \mathrm { AB } } | = 8 , \overrightarrow { \mathrm { AC } } \cdot \overrightarrow { \mathrm { BC } } = 0$ (나) $\overrightarrow { \mathrm { AD } } = \frac { 1 } { 2 } \overrightarrow { \mathrm { AB } } - 2 \overrightarrow { \mathrm { BC } }$
(1) 32
(2) 34
(3) 36
(4) 38
(5) 40

For two vectors $\vec{a}$ and $\vec{b}$, $$|\vec{a}| = \sqrt{11}, \quad |\vec{b}| = 3, \quad |2\vec{a} - \vec{b}| = \sqrt{17}$$ What is the value of $|\vec{a} - \vec{b}|$? [3 points]
(1) $\frac{\sqrt{2}}{2}$
(2) $\sqrt{2}$
(3) $\frac{3\sqrt{2}}{2}$
(4) $2\sqrt{2}$
(5) $\frac{5\sqrt{2}}{2}$

13. Given that $\vec { e } _ { 1 } , \vec { e } _ { 2 }$ are unit vectors in the plane, and $\vec { e } _ { 1 } \cdot \vec { e } _ { 2 } = \frac { 1 } { 2 }$ . If the plane vector $\vec { b }$ satisfies $\vec { b } \cdot \vec { e } _ { 1 } = \vec { b } \cdot \vec { e } _ { 2 } = 1$ , then $| \vec { b } | =$ $\_\_\_\_$.

Given vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ with an angle of $60 ^ { \circ }$ between them, $| \boldsymbol{a} | = 2$, and $| \boldsymbol{b} | = 1$, then $| \boldsymbol{a} + 2 \boldsymbol{b} | = $ \_\_\_\_

13. If vectors $\vec { a } , \vec { b }$ satisfy $| \vec { a } | = 3 , | \vec { a } - \vec { b } | = 5 , \vec { a } \cdot \vec { b } = 1$, then $| \vec { b } | =$ $\_\_\_\_$ .

Given vectors $a = ( 2,1 ) , b = ( - 2,4 )$ , then $| a - b | =$
A. 2
B. 3
C. 4
D. 5

Given vectors $\vec { a } , \vec { b }$ satisfying $| \vec { a } | = 1 , | \vec { a } + 2 \vec { b } | = 2$, and $( \vec { b } - 2 \vec { a } ) \perp \vec { b }$, then $| \vec { b } | =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 2 } } { 2 }$
C. $\frac { \sqrt { 3 } } { 2 }$
D. 1

142- Three vectors $\vec{a} = (1,1,0)$, $\vec{b} = (-1,2,0)$, and $\vec{c}$ are non-coplanar, and $\vec{h} = (x,y,4)$ is the altitude vector of the parallelepiped formed by these three vectors. If $\vec{a} \cdot \vec{c} = 1$ and $\vec{b} \cdot \vec{c} = 5$, what is the magnitude of vector $\vec{c}$?
(4) $\sqrt{21}$ (3) $\sqrt{19}$ (2) $4$ (1) $5$

24. Let A0 A1 A2 A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A0 A1 A0 and A0 A4 is :
(A) $3 / 4$
(B) $3 \sqrt { } 3$
(C) 3
(D) $( 3 \sqrt { } 3 ) / 2$

10. Let $\vec { a } = 2 \hat { \imath } + \hat { \jmath } - 2 \hat { k }$ and $\hat { b } = \hat { \imath } + \hat { \jmath }$. If $\hat { c }$ is a vector such that $\vec { a } \cdot \vec { c } = | \vec { c } - \vec { a } | = 2 \sqrt { 2 }$ and the angle between $( \vec { a } \times \vec { b } )$ and $\vec { c }$ is $30 ^ { \circ }$, then $| ( \vec { a } \times \vec { b } ) \times \vec { c } | =$
(A) $\frac { 3 } { 2 }$
(B) $\frac { 3 } { 2 }$
(C) 2
(D) 3

27. Let $\vec { a }$ and $\vec { b }$ be two non-collinear unit vectors. If $\vec { u } = \vec { a } - ( \vec { a } , \vec { b } ) \vec { b }$ and $\vec { v } = \vec { a } \times \vec { b }$, then $| \vec { v } |$ is:
(A) $| \vec { u } |$
(B) $\quad | \vec { u } | + | \vec { u } \cdot \vec { a } |$
(C) $| \vec { u } | + | \vec { u } , \vec { b } |$
(D) $| \vec { u } | + \vec { u } \cdot ( \vec { a } + \vec { b } )$

The resultant of two forces P N and 3 N is a force of 7 N . If the direction of 3 N force were reversed, the resultant would be $\sqrt { 19 } \mathrm {~N}$. The value of P is
(1) 5 N
(2) 6 N
(3) 3 N
(4) 4 N

If the vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a triangle $ABC$, then the length of the median through $A$ is:
(1) $\sqrt{33}$
(2) $\sqrt{45}$
(3) $\sqrt{18}$
(4) $\sqrt{72}$

If $| \vec { a } | = 2 , | \vec { b } | = 3$ and $| \overrightarrow { 2 a } - \vec { b } | = 5$, then $| \overrightarrow { 2 a } + \vec { b } |$ equals:
(1) 5
(2) 7
(3) 17
(4) 1

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 9$, then $|2\vec{a} + 5\vec{b} + 5\vec{c}|$ is:
(1) $3$
(2) $\sqrt{10}$
(3) $2$
(4) $\sqrt{5}$

Let the vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ be such that $|\overrightarrow{\mathrm{a}}| = 2$, $|\overrightarrow{\mathrm{b}}| = 4$ and $|\overrightarrow{\mathrm{c}}| = 4$. If the projection of $\overrightarrow{\mathrm{b}}$ on $\overrightarrow{\mathrm{a}}$ is equal to the projection of $\overrightarrow{\mathrm{c}}$ on $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is perpendicular to $\overrightarrow{\mathrm{c}}$, then the value of $|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}|$ is ...

Let $\vec { x }$ be a vector in the plane containing vectors $\vec { a } = 2 \hat { i } - \hat { j } + \hat { k }$ and $\vec { b } = \hat { i } + 2 \hat { j } - \hat { k }$. If the vector $\vec { x }$ is perpendicular to $( 3 \hat { i } + 2 \hat { j } - \widehat { k } )$ and its projection on $\vec { a }$ is $\frac { 17 \sqrt { 6 } } { 2 }$, then the value of $| \vec { x } | ^ { 2 }$ is equal to $\_\_\_\_$ .

Let $\vec { a }$ and $\vec { b }$ be two vectors such that $| 2 \vec { a } + 3 \vec { b } | = | 3 \vec { a } + \vec { b } |$ and the angle between $\vec { a }$ and $\vec { b }$ is $60 ^ { \circ }$. If $\frac { 1 } { 8 } \vec { a }$ is a unit vector, then $| \vec { b } |$ is equal to :
(1) 8
(2) 4
(3) 6
(4) 5

Two bodies of mass 1 kg and 3 kg have position vectors $\hat { \mathrm { i } } + 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ and $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :
(1) $\hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$
(2) $- 3 \hat { \mathrm { i } } - 2 \hat { \mathrm { j } } + \widehat { \mathrm { k } }$
(3) $- 2 \hat { \mathrm { i } } + 2 \widehat { \mathrm { k } }$
(4) $- 2 \hat { \mathrm { i } } - \hat { \mathrm { j } } + 2 \widehat { \mathrm { k } }$