Vector

Problem 1

已知向量 $\vec{a}, \vec{b}$ 满足 $|\vec{a}|=1,|\vec{b}|=\sqrt{3},|\vec{a}-2 \vec{b}|=3$ ，求 $\vec{a} \cdot \vec{b}$ 。

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Problem 2

Find the reflection of the point $P=(-4,-1)$ across the x-axis. What are the new coordinates?

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Problem 3

Find the new coordinates of $(-1,3)$ after moving 2 units down and rotating $270^{\circ}$ counterclockwise around the origin.

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Problem 4

Find the image of point $P=(2,-1)$ after a $270^{\circ}$ counterclockwise rotation about the origin.

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Problem 5

Four points A, B, C, D on a semicircle satisfy $|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BC}}|=|\overrightarrow{\mathrm{CD}}|$ . Assess the validity of assertions A and R.

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Problem 6

¿Cuáles son magnitudes vectoriales? a. desplazamiento, trabajo, energía b. fuerza, aceleración, temperatura c. aceleración, masa, tiempo d. velocidad, desplazamiento, fuerza

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Problem 7

Un vector tiene magnitud y dirección, mientras que un escalar solo tiene magnitud. ¿Cuál es la diferencia?

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Problem 8

Calcula la magnitud de un vector con $x=3$ y $y=2$ . Opciones: a. 5 b. 3.6 c. 6 d. 1

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Problem 9

La magnitud de un vector unitario es igual a: a. mayor a uno. b. $\sqrt{\text{suma de componentes al cuadrado}}$ . c. múltiplo de la magnitud del vector desplazamiento. d. igual a 1.

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Problem 10

Three charges of $+6.65 \mu \mathrm{C}$ at the triangle's vertices create a force. Find the resultant force: a. $172.34 \mathrm{~N}$ b. $22.98 \mathrm{~N}$ c. $34.47 \mathrm{~N}$ d. $68.94 \mathrm{~N}$ e. $0.89 \mathrm{~N}$

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Problem 11

Determine the location of the point $(-3,4)$ in the coordinate plane.

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Problem 12

Evaluate $\int F \cdot dr$ for $\bar{f}=x y \bar{i}-z \hat{j}+x^{2} \bar{k}$ along $\alpha: x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ . Ans: $\frac{51}{70}$

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Problem 13

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ .

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Problem 14

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}$ , $y=2t$ , $z=t^{3}$ from $t=0$ to $t=1$ . Answer: $\frac{51}{70}$ .

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Problem 15

Calculate the work done by the force $\bar{f}=\bar{z} \bar{i}+x \bar{j}+y \hat{k}$ along the curve $\bar{r}=\cos t i+\sin t j-t \bar{k}$ from $t=0$ to $t=2 \pi$ .

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Problem 16

Evaluate $\oint_{C}(y z \, dl + x z \, dy + l y \, dz)$ for the helix $x=a \cos t, y=a \sin t, z=k t$ as $t$ goes from 0 to $2\pi$ .

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Problem 17

Translate Triangle RST with vertices $R (-4,2)$ , $S (5,3)$ , $T (2,-5)$ , 4 down and 3 left. Find new coordinates.

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Problem 18

Find point $B$ on line segment $AC$ where $A(6,-6)$ , $C(-6,-2)$ , and $AB=\frac{3}{4}AC$ .

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Problem 19

Gegeben sind die Punkte A(2,-3,0), B(2,2,0), C(-1,2,0) und E(2,-3,5) eines Quaders. Finde die anderen Eckpunkte und berechne die Diagonale $\overline{A G}$ .

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Problem 20

Berechnen Sie die Vektor-Koordinaten aus den folgenden Linearkombinationen: a) $\left(\begin{array}{l}2 \\ 1 \\ 3\end{array}\right)+\left(\begin{array}{r}1 \\ 4 \\ -5\end{array}\right)+2 \cdot\left(\begin{array}{l}1 \\ 1 \\ 4\end{array}\right)$ b) $3 \cdot\left(\begin{array}{r}2 \\ 10 \\ 0\end{array}\right)+2 \cdot\left(\begin{array}{r}0 \\ 3 \\ -1\end{array}\right)-5 \cdot\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)$ c) $\frac{1}{2} \cdot\left(\begin{array}{l}3 \\ 1 \\ 2\end{array}\right)+\frac{1}{3} \cdot\left(\begin{array}{r}1 \\ 2 \\ -2\end{array}\right)-\frac{1}{4} \cdot\left(\begin{array}{l}2 \\ 2 \\ 4\end{array}\right)$

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Problem 21

Gegeben sind die Punkte A(2|-3|0), B(2|2|0), C(-1|2|0) und E(2|-3|5) eines Quaders. Finde die anderen Eckpunkte und die Länge der Diagonale $\overline{A G}$ .

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Problem 22

Find the projection of the vector $u=3 i+j+k$ onto the vector $a=4 j-3 k$ .

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Problem 23

Find the component of the vector $a=3i+j-k$ in the direction of $b=i-2j+6k$ .

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Problem 24

Find the new vertices $K^{\prime}, L^{\prime}, M^{\prime}, N^{\prime}$ after rotating polygon KLMN by $90^{\circ}$ clockwise.

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Problem 25

Given lines $O A N, O M B$ and $A P B$ , with $A N = 2 O A$ . If $M$ is the midpoint of $O B$ , find $k$ in $\overrightarrow{A P} = k \overrightarrow{A B}$ , given $MPN$ is straight.

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Problem 26

Describe the symbols: alou $\overleftrightarrow{P Q}$ , $\overrightarrow{P Q}$ , $\overrightarrow{P Q}$ , $\overrightarrow{Q P}$ .

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Problem 27

Find the midpoint $M$ of the line segment with endpoints $G(5,4)$ and $H(7,4)$ . Write $M$ as decimals or integers. $M=$

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Problem 28

Rotate the vector $(-5,3)$ using the matrix $\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]$ . Find the terminal $x$ -value. $x=$

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Problem 29

Rotate the vector $\begin{bmatrix}-5 \\ -3\end{bmatrix}$ using the matrix $\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$ . Find the $x$ -value of the new vector. $x=$

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Problem 30

John cycles south 15 m, east 21 m, then south 5 m to point Y. Calculate the distance from X to Y.

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Problem 31

Find the distance and bearing from Corpus Christi to the Coast Guard cutter after it travels at 30 knots for 4 hours on given courses.

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Problem 32

A boat travels at 40 knots on a course of $65^{\circ}$ for 2 hours, then $155^{\circ}$ for 4 hours. Find the distance and bearing from Fort Lauderdale.

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Problem 33

Find the vertices of parallelogram $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} D^{\prime \prime}$ after dilation by 4 and reflection. Are the parallelograms similar or congruent?

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Problem 34

Find the midpoint of the segment with endpoints $(5,11)$ and $(3,1)$ .

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Problem 35

Find the midpoint of the line segment between the points $(-2,4)$ and $(2,6)$ .

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Problem 36

Find the midpoint of the segment between the points $(-7,5)$ and $(7,7)$ .

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Problem 37

Sketch an angle $\theta$ with the point $(-6,0)$ on its terminal side. Find the six trig functions of $\theta$ .

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Problem 38

Find the new coordinates of B (-5,-8), C (-5,-3), D (0,-3), E (0,-8) after a $180^{\circ}$ rotation around the origin.

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Problem 39

Find the new coordinates of vertices B $(2,-9)$ , C $(2,-4)$ , and D $(1,-9)$ after a $270^{\circ}$ clockwise rotation.

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Problem 40

A train starts from rest. After 40 s, it reaches 60 m/s. What is its acceleration?

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Problem 41

After 4 seconds, where is the sprite located? Choose from: (25,-25), (-25,0), (0,25), (-25,25).

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Problem 42

Sketch the figure for $\ddot{XY} \perp \overrightarrow{YZ}$ . What’s the first step: A, B, C, or D?

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Problem 43

Find point $B$ on line segment $\overline{AC}$ such that the ratio $AB : AC = 1 : 3$ where $A=(-2,4)$ and $C=(4,7)$ .

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Problem 44

1. Find the midpoint of $\overline{B D}$ where B is at $(-5,-6)$ and D is at $(9,11)$ .
2. If M is the midpoint of $\overline{X Y}$ with $X(1,-2)$ and $M(7,4)$ , find Y's coordinates.
3. Calculate the length of $\overline{R S}$ for $R(8,2)$ and $S(3,7)$ , rounding to the nearest whole number.
4. Find the weighted average of 6 (weight 4) and 12 (weight 2).

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Problem 45

Which coordinates for points $A^{\prime}$ and $B^{\prime}$ show that lines $AB$ and $A^{\prime}B^{\prime}$ are perpendicular?
1. $A^{\prime}:(p, m)$ and $B^{\prime}:(z, w)$
2. $A^{\prime}:(p, m)$ and $B^{\prime}:(z,-w)$
3. $A^{\prime}:(p,-m)$ and $B^{\prime}:(z, w)$
4. $A^{\prime}:(p,-m)$ and $B^{\prime}:(z,-w)$

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Problem 46

Find the pre-image of vertex $A^{\prime}(8,5)$ after reflection across the $y$ -axis. Options: $(-8,-6)$ , $(-6,8)$ , $(8,6)$ , $(6,-8)$ .

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Problem 47

Find which reflection of the segment from $(-4,-6)$ to $(-6,4)$ gives endpoints $(4,-6)$ and $(6,4)$ .

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Problem 48

Combine the following translations into a single translation:
21. $T_{\langle-3,3\rangle} \circ T_{\langle-2,4\rangle}$
22. $T_{\langle-4,-3\rangle} \circ T_{\langle 3,1\rangle}$
23. $T_{\langle 5,-6\rangle} \circ T_{\langle-7,5\rangle}$
24. $T_{\langle 8,-2\rangle} \circ T_{\langle-4,9\rangle}$

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Problem 49

Find the midpoint of the line segment with endpoints $K(7,7)$ and $L(1,3)$ . Choose from the options: A. $(-3,-2)$ B. $(3,-2)$ C. $(-3,2)$ D. $(3,2)$ .

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Problem 50

Find the midpoint of the segment with endpoints $(-2,3)$ and $(5,-3)$ .

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Problem 51

Identify the opposite ray to $\overrightarrow{N M}$ from the options: $\overrightarrow{N O}$ , $M$ , $\overrightarrow{N P}$ , $\overrightarrow{M P}$ , or all of the above.

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Problem 52

Find the midpoint of the segment with endpoints $(-9,-2)$ and $(-1,-8)$ .

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Problem 53

Find the coordinates that divide the line segment from $(2,-6)$ to $(5,6)$ in a 1:5 ratio.

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Problem 54

Find the area of the parallelogram with vertices at (-4,-5), (3,-3), (-4,-9), (3,-7).

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Problem 55

Graph the reflection of the point $T(10,-2)$ over the $x$ -axis.

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Problem 56

Reflect the point $S(0,2)$ over the $x$ -axis. What are the coordinates of $S^{\prime}$ ?

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Problem 57

Find $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-3 \mathbf{v}$ for $u=\begin{bmatrix}-3 \\ 6\end{bmatrix}, v=\begin{bmatrix}-5 \\ 4\end{bmatrix}$ .

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Problem 58

Find $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-3 \mathbf{v}$ for $u=\begin{bmatrix}-3 \\ 6\end{bmatrix}$ and $v=\begin{bmatrix}-5 \\ 4\end{bmatrix}$ .

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Problem 59

Is vector $b$ a linear combination of $a_{1}, a_{2}, a_{3}$ ? Choose A, B, C, or D based on the echelon matrix pivots.

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Problem 60

Describe the Span $\{\mathbf{v}_{1}, \mathbf{v}_{2}\}$ for $\mathbf{v}_{1}=\begin{bmatrix}4 \\ 10 \\ -2\end{bmatrix}$ and $\mathbf{v}_{2}=\begin{bmatrix}10 \\ 25 \\ -5\end{bmatrix}$ . Choose A, B, C, or D.

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Problem 61

Find the value(s) of $h$ so that the vector $b = \left[\begin{array}{r}4 \\ -9 \\ h\end{array}\right]$ lies in the plane spanned by $a_{1} = \left[\begin{array}{r}1 \\ 3 \\ -1\end{array}\right]$ and $a_{2} = \left[\begin{array}{r}-6 \\ -11 \\ 2\end{array}\right]$ . The value(s) of $h$ is(are) $\square$ .

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Problem 62

Do the vectors $\mathbf{v}_{1} = \begin{bmatrix} 0 \\ 0 \\ -3 \end{bmatrix}$ , $\mathbf{v}_{2} = \begin{bmatrix} 0 \\ -5 \\ 6 \end{bmatrix}$ , and $\mathbf{v}_{3} = \begin{bmatrix} 5 \\ -3 \\ 9 \end{bmatrix}$ span $\mathbb{R}^{3$? Explain.

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Problem 63

Find the vectors $\overrightarrow{P Q}$ and $\overrightarrow{P R}$ for points $P(-2,1)$ , $Q(5,3)$ , and $R(x,y)$ .

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Problem 64

Find the value(s) of $h$ such that the vector $b=\left[\begin{array}{l}5 \\ 3 \\ h\end{array}\right]$ lies in the plane spanned by $a_{1}=\left[\begin{array}{r}1 \\ 3 \\ -1\end{array}\right]$ and $a_{2}=\left[\begin{array}{r}-5 \\ -11 \\ 2\end{array}\right]$ . The value(s) of $h$ is(are) $\square$ .

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Problem 65

Do the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$ span $\mathbb{R}^{4}$ ? Choose A, B, C, or D for the answer.

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Problem 66

Find the midpoint of the segment with endpoints A(1.8,-4.3) and B(-5.6,-6.5). Options: (-3.8,-10.8), (-1.9,-5.4), (3.7,5.4), (7.4,10.8).

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Problem 67

Translate the point (1,-6) by 2 units right and 6 units down. Show your work.

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Problem 68

Find the translation from point $Q(-9,-5)$ to $Q^{\prime}(-2,-8)$ as $\langle x, y \rangle$ .

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Problem 69

Find the translation from point $Q(-9,-5)$ to $Q^{\prime}(-2,-8)$ as $\langle x, y\rangle$ . Show your work.

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Problem 70

Find the translation from $Q(3,6)$ to $Q^{\prime}(9,3)$ as $\langle x, y\rangle$ . Show your work.

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Problem 71

A ship moves at 30 knots on a 64° bearing for 2 hours, then turns to 154° for 3 hours. Find the distance and bearing from start.

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Problem 72

Find the unit vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ .

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Problem 73

Find the distance, midpoint, and slope of the line between points $(2,-2)$ and $(5,2)$ .

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Problem 74

Translate the vertices A(2,-6), B(-1,-1), C(-3,-5) by $(x, y) \rightarrow (x+3, y+5)$ . Find A', B', C'.

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Problem 75

A translation moves point $V(-2,3)$ to $V^{\prime}(-2,7$ . Identify true statements about the translation.

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Problem 76

Find new coordinates of vertices $M'$ , $P'$ , $Q'$ , and $V'$ after a $270^{\circ}$ rotation of parallelogram $MPQV$ around $(-5,-10)$ .

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Problem 77

Find the midpoint of the segment from $(-1,5)$ to $(-2,3)$ . What is Midpoint = ?

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Problem 78

A boat goes N 36° 40' W for 62.5 miles. Find north and west distances traveled, rounded to 0.1 miles.

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Problem 79

A boat sails N 38° 10' W for 78.3 miles. Find the north and west distances traveled, rounded to the nearest tenth.

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Problem 80

Find the coordinates of point $Y$ that divides segment $XZ$ ( $X(-4,3)$ , $Z(6,-2)$ ) one-fifth from $X$ to $Z$ .

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Problem 81

Find the new vertices of $\triangle A B C$ after these translations: 1. $T_{\langle-2,3\rangle}$ , 2. $T_{\langle-4,-1\rangle}$ , 3. $T_{(4,6)}$ .

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Problem 82

Find the new coordinates of point $C'$ after rotating point $C(2, -3)$ 90 degrees clockwise and translating left by 2 units. Options: $(-1,2)$ , $(-5,2)$ , $(-6,-3)$ , $(-3,2)$ .

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Problem 83

Find the vector $\mathbf{F}$ from the equations:
1500x + 5000y + z = 1300, 3200x + 12000y + z = 5300, 4300x + 13000y + z = 6500.
$\mathbf{F}$ is the constants: [1300, 5300, 6500].

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Problem 84

Translate points D(-4,-5), E(0,-5), F(-1,-3), G(-3,-3) left 3 units and down 2 units. Find D', E', F', G'.

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Problem 85

Rotate points $U(-3,6)$ , $V(-8,1)$ , and $W(-3,1)$ by $180^{\circ}$ around the origin. Find $U^{\prime}, V^{\prime}, W^{\prime}$ .

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Problem 86

Describe the rotation that transforms triangle DEF with vertices $D(0,3), E(1,8), F(-3,4)$ to $D^{\prime}(3,0), E^{\prime}(8,-1), F(4,3)$ .

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Problem 87

Find the translation from triangle $K M N$ with vertices $K(12,3), M(-5,2), N(8,-4)$ to $K^{\prime}(18,0), M^{\prime}(1,-1), N^{\prime}(14,-7)$ .

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Problem 88

Describe the rotation that transforms triangle $P Q R$ with vertices $P(9,-2)$ , $Q(1,0)$ , $R(-7,3)$ to $P^{\prime}(-9,2)$ , $Q^{\prime}(-1,0)$ , $R^{\prime}(7,-3)$ .

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Problem 89

Find the rotation that transforms triangle $PQR$ with vertices $P(9,-2)$ , $Q(1,0)$ , $R(-7,3)$ to $P'(-9,2)$ , $Q'(-1,0)$ , $R'(7,-3)$ .

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Problem 90

Reflect triangle STW across the x-axis to get triangle S'T'W' with vertices $S'(15,6)$ , $T'(-2,-3)$ , $W'(-8,8)$ .

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Problem 91

Sketch the polar point (-6, -25π/12).

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Problem 92

Finde die Koordinaten der beiden anderen Ecken eines Quadrats mit einer Seitenkante $AB$ zwischen $A(3 \mid 4)$ und $B(7 \mid 6)$ .

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Problem 93

Find the point that is $\frac{7}{10}$ of the way from $A(-3,-5)$ to $B(9,7)$ .

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Problem 94

Calculate Michael Phelps' resultant velocity swimming North at $14 \mathrm{~m/s}$ against a $5 \mathrm{~m/s}$ South current.

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Problem 95

Find the resultant velocity when crossing a river with a current of $15 \mathrm{~m/s}$ downstream and crossing at $2 \mathrm{~m/s}$ north.

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Problem 96

Find slopes and lengths for quadrilateral with points Q(1,3), R(3,-4), S(9,-7), T(7,0). Identify its type.

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Problem 97

Find the point that is $\frac{3}{10}$ of the way from $A(-3,-6)$ to $B(10,5)$ .

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Problem 98

Plot the point (3, -\frac{7 \pi}{4}).

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Problem 99

Two ships leave a port. One sails at $48^{\circ}$ , 12 knots; the other at $138^{\circ}$ , 22 knots. Distance apart after 1.5 hours?

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Problem 100

Two ships leave a port at the same time. After 1.5 hours, how far apart are they if one sails at 24 knots and the other at 26 knots?

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Vector

Problem 1

已知向量 a⃗,b⃗ \vec{a}, \vec{b} a,b 满足 ∣a⃗∣=1,∣b⃗∣=3,∣a⃗−2b⃗∣=3 |\vec{a}|=1,|\vec{b}|=\sqrt{3},|\vec{a}-2 \vec{b}|=3 ∣a∣=1,∣b∣=3​,∣a−2b∣=3，求 a⃗⋅b⃗ \vec{a} \cdot \vec{b} a⋅b。

Problem 2

Find the reflection of the point P=(−4,−1)P=(-4,-1)P=(−4,−1) across the x-axis. What are the new coordinates?

Problem 3

Find the new coordinates of (−1,3)(-1,3)(−1,3) after moving 2 units down and rotating 270∘270^{\circ}270∘ counterclockwise around the origin.

Problem 4

Find the image of point P=(2,−1)P=(2,-1)P=(2,−1) after a 270∘270^{\circ}270∘ counterclockwise rotation about the origin.

Problem 5

Four points A, B, C, D on a semicircle satisfy ∣ABundefined∣=∣BCundefined∣=∣CDundefined∣|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BC}}|=|\overrightarrow{\mathrm{CD}}|∣AB∣=∣BC∣=∣CD∣. Assess the validity of assertions A and R.

Problem 6

¿Cuáles son magnitudes vectoriales? a. desplazamiento, trabajo, energía b. fuerza, aceleración, temperatura c. aceleración, masa, tiempo d. velocidad, desplazamiento, fuerza

Problem 7

Un vector tiene magnitud y dirección, mientras que un escalar solo tiene magnitud. ¿Cuál es la diferencia?

Problem 8

Calcula la magnitud de un vector con x=3x=3x=3 y y=2y=2y=2. Opciones: a. 5 b. 3.6 c. 6 d. 1

Problem 9

La magnitud de un vector unitario es igual a: a. mayor a uno. b. suma de componentes al cuadrado\sqrt{\text{suma de componentes al cuadrado}}suma de componentes al cuadrado​. c. múltiplo de la magnitud del vector desplazamiento. d. igual a 1.

Problem 10

Three charges of +6.65μC+6.65 \mu \mathrm{C}+6.65μC at the triangle's vertices create a force. Find the resultant force: a. 172.34 N172.34 \mathrm{~N}172.34 N b. 22.98 N22.98 \mathrm{~N}22.98 N c. 34.47 N34.47 \mathrm{~N}34.47 N d. 68.94 N68.94 \mathrm{~N}68.94 N e. 0.89 N0.89 \mathrm{~N}0.89 N

Problem 11

Determine the location of the point (−3,4)(-3,4)(−3,4) in the coordinate plane.

Problem 12

Evaluate ∫F⋅dr\int F \cdot dr∫F⋅dr for fˉ=xyiˉ−zj^+x2kˉ\bar{f}=x y \bar{i}-z \hat{j}+x^{2} \bar{k}fˉ​=xyiˉ−zj^​+x2kˉ along α:x=t2,y=2t,z=t3\alpha: x=t^{2}, y=2t, z=t^{3}α:x=t2,y=2t,z=t3 from t=0t=0t=0 to t=1t=1t=1. Ans: 5170\frac{51}{70}7051​

Problem 13

Evaluate ∫F⋅dr\int F \cdot d r∫F⋅dr for Fˉ=xyiˉ−zj+x2kˉ\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}Fˉ=xyiˉ−zj+x2kˉ along the curve x=t2,y=2t,z=t3x=t^{2}, y=2t, z=t^{3}x=t2,y=2t,z=t3 from t=0t=0t=0 to t=1t=1t=1.

Problem 14

Evaluate ∫F⋅dr\int F \cdot d r∫F⋅dr for Fˉ=xyiˉ−zj+x2kˉ\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}Fˉ=xyiˉ−zj+x2kˉ along the curve x=t2x=t^{2}x=t2, y=2ty=2ty=2t, z=t3z=t^{3}z=t3 from t=0t=0t=0 to t=1t=1t=1. Answer: 5170\frac{51}{70}7051​.

Problem 15

Calculate the work done by the force fˉ=zˉiˉ+xjˉ+yk^\bar{f}=\bar{z} \bar{i}+x \bar{j}+y \hat{k}fˉ​=zˉiˉ+xjˉ​+yk^ along the curve rˉ=cos⁡ti+sin⁡tj−tkˉ\bar{r}=\cos t i+\sin t j-t \bar{k}rˉ=costi+sintj−tkˉ from t=0t=0t=0 to t=2πt=2 \pit=2π.

Problem 16

Evaluate ∮C(yz dl+xz dy+ly dz)\oint_{C}(y z \, dl + x z \, dy + l y \, dz)∮C​(yzdl+xzdy+lydz) for the helix x=acos⁡t,y=asin⁡t,z=ktx=a \cos t, y=a \sin t, z=k tx=acost,y=asint,z=kt as ttt goes from 0 to 2π2\pi2π.

Problem 17

Translate Triangle RST with vertices R(−4,2)R (-4,2)R(−4,2), S(5,3)S (5,3)S(5,3), T(2,−5)T (2,-5)T(2,−5), 4 down and 3 left. Find new coordinates.

Problem 18

Find point BBB on line segment ACACAC where A(6,−6)A(6,-6)A(6,−6), C(−6,−2)C(-6,-2)C(−6,−2), and AB=34ACAB=\frac{3}{4}ACAB=43​AC.

Problem 19

Gegeben sind die Punkte A(2,-3,0), B(2,2,0), C(-1,2,0) und E(2,-3,5) eines Quaders. Finde die anderen Eckpunkte und berechne die Diagonale AG‾\overline{A G}AG.

Problem 20

Problem 21

Gegeben sind die Punkte A(2|-3|0), B(2|2|0), C(-1|2|0) und E(2|-3|5) eines Quaders. Finde die anderen Eckpunkte und die Länge der Diagonale AG‾\overline{A G}AG.

Problem 22

Find the projection of the vector u=3i+j+ku=3 i+j+ku=3i+j+k onto the vector a=4j−3ka=4 j-3 ka=4j−3k.

Problem 23

Find the component of the vector a=3i+j−ka=3i+j-ka=3i+j−k in the direction of b=i−2j+6kb=i-2j+6kb=i−2j+6k.

Problem 24

Find the new vertices K′,L′,M′,N′K^{\prime}, L^{\prime}, M^{\prime}, N^{\prime}K′,L′,M′,N′ after rotating polygon KLMN by 90∘90^{\circ}90∘ clockwise.

Problem 25

Given lines OAN,OMBO A N, O M BOAN,OMB and APBA P BAPB, with AN=2OAA N = 2 O AAN=2OA. If MMM is the midpoint of OBO BOB, find kkk in APundefined=kABundefined\overrightarrow{A P} = k \overrightarrow{A B}AP=kAB, given MPNMPNMPN is straight.

Problem 26

Describe the symbols: alou PQundefined\overleftrightarrow{P Q}PQ​, PQundefined\overrightarrow{P Q}PQ​, PQundefined\overrightarrow{P Q}PQ​, QPundefined\overrightarrow{Q P}QP​.

Problem 27

Find the midpoint MMM of the line segment with endpoints G(5,4)G(5,4)G(5,4) and H(7,4)H(7,4)H(7,4). Write MMM as decimals or integers. M=M= M=

Problem 28

Rotate the vector (−5,3)(-5,3)(−5,3) using the matrix [01−10]\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right][0−1​10​]. Find the terminal xxx-value. x= x= x=

Problem 29

Rotate the vector [−5−3]\begin{bmatrix}-5 \\ -3\end{bmatrix}[−5−3​] using the matrix [01−10]\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}[0−1​10​]. Find the xxx-value of the new vector. x= x= x=

Problem 30

John cycles south 15 m, east 21 m, then south 5 m to point Y. Calculate the distance from X to Y.

Problem 31

Find the distance and bearing from Corpus Christi to the Coast Guard cutter after it travels at 30 knots for 4 hours on given courses.

Problem 32

A boat travels at 40 knots on a course of 65∘65^{\circ}65∘ for 2 hours, then 155∘155^{\circ}155∘ for 4 hours. Find the distance and bearing from Fort Lauderdale.

Problem 33

Find the vertices of parallelogram A′′B′′C′′D′′A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} D^{\prime \prime}A′′B′′C′′D′′ after dilation by 4 and reflection. Are the parallelograms similar or congruent?

Problem 34

Find the midpoint of the segment with endpoints (5,11)(5,11)(5,11) and (3,1)(3,1)(3,1).

Problem 35

Find the midpoint of the line segment between the points (−2,4)(-2,4)(−2,4) and (2,6)(2,6)(2,6).

Problem 36

Find the midpoint of the segment between the points (−7,5)(-7,5)(−7,5) and (7,7)(7,7)(7,7).

Problem 37

Sketch an angle θ\thetaθ with the point (−6,0)(-6,0)(−6,0) on its terminal side. Find the six trig functions of θ\thetaθ.

Problem 38

Find the new coordinates of B (-5,-8), C (-5,-3), D (0,-3), E (0,-8) after a 180∘180^{\circ}180∘ rotation around the origin.

Problem 39

Find the new coordinates of vertices B (2,−9)(2,-9)(2,−9), C (2,−4)(2,-4)(2,−4), and D (1,−9)(1,-9)(1,−9) after a 270∘270^{\circ}270∘ clockwise rotation.

Problem 40

A train starts from rest. After 40 s, it reaches 60 m/s. What is its acceleration?

已知向量 $\vec{a}, \vec{b}$ 满足 $|\vec{a}|=1,|\vec{b}|=\sqrt{3},|\vec{a}-2 \vec{b}|=3$ ，求 $\vec{a} \cdot \vec{b}$ 。

Find the reflection of the point $P=(-4,-1)$ across the x-axis. What are the new coordinates?

Find the new coordinates of $(-1,3)$ after moving 2 units down and rotating $270^{\circ}$ counterclockwise around the origin.

Find the image of point $P=(2,-1)$ after a $270^{\circ}$ counterclockwise rotation about the origin.

Four points A, B, C, D on a semicircle satisfy $|\overrightarrow{\mathrm{AB}}|=|\overrightarrow{\mathrm{BC}}|=|\overrightarrow{\mathrm{CD}}|$ . Assess the validity of assertions A and R.

Calcula la magnitud de un vector con $x=3$ y $y=2$ . Opciones: a. 5 b. 3.6 c. 6 d. 1

La magnitud de un vector unitario es igual a: a. mayor a uno. b. $\sqrt{\text{suma de componentes al cuadrado}}$ . c. múltiplo de la magnitud del vector desplazamiento. d. igual a 1.

Three charges of $+6.65 \mu \mathrm{C}$ at the triangle's vertices create a force. Find the resultant force: a. $172.34 \mathrm{~N}$ b. $22.98 \mathrm{~N}$ c. $34.47 \mathrm{~N}$ d. $68.94 \mathrm{~N}$ e. $0.89 \mathrm{~N}$

Determine the location of the point $(-3,4)$ in the coordinate plane.

Evaluate $\int F \cdot dr$ for $\bar{f}=x y \bar{i}-z \hat{j}+x^{2} \bar{k}$ along $\alpha: x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ . Ans: $\frac{51}{70}$

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}, y=2t, z=t^{3}$ from $t=0$ to $t=1$ .

Evaluate $\int F \cdot d r$ for $\bar{F}=x y \bar{i}-z j+x^{2} \bar{k}$ along the curve $x=t^{2}$ , $y=2t$ , $z=t^{3}$ from $t=0$ to $t=1$ . Answer: $\frac{51}{70}$ .

Calculate the work done by the force $\bar{f}=\bar{z} \bar{i}+x \bar{j}+y \hat{k}$ along the curve $\bar{r}=\cos t i+\sin t j-t \bar{k}$ from $t=0$ to $t=2 \pi$ .

Evaluate $\oint_{C}(y z \, dl + x z \, dy + l y \, dz)$ for the helix $x=a \cos t, y=a \sin t, z=k t$ as $t$ goes from 0 to $2\pi$ .

Translate Triangle RST with vertices $R (-4,2)$ , $S (5,3)$ , $T (2,-5)$ , 4 down and 3 left. Find new coordinates.

Find point $B$ on line segment $AC$ where $A(6,-6)$ , $C(-6,-2)$ , and $AB=\frac{3}{4}AC$ .

Gegeben sind die Punkte A(2,-3,0), B(2,2,0), C(-1,2,0) und E(2,-3,5) eines Quaders. Finde die anderen Eckpunkte und berechne die Diagonale $\overline{A G}$ .

Gegeben sind die Punkte A(2|-3|0), B(2|2|0), C(-1|2|0) und E(2|-3|5) eines Quaders. Finde die anderen Eckpunkte und die Länge der Diagonale $\overline{A G}$ .

Find the projection of the vector $u=3 i+j+k$ onto the vector $a=4 j-3 k$ .

Find the component of the vector $a=3i+j-k$ in the direction of $b=i-2j+6k$ .

Find the new vertices $K^{\prime}, L^{\prime}, M^{\prime}, N^{\prime}$ after rotating polygon KLMN by $90^{\circ}$ clockwise.

Given lines $O A N, O M B$ and $A P B$ , with $A N = 2 O A$ . If $M$ is the midpoint of $O B$ , find $k$ in $\overrightarrow{A P} = k \overrightarrow{A B}$ , given $MPN$ is straight.

Describe the symbols: alou $\overleftrightarrow{P Q}$ , $\overrightarrow{P Q}$ , $\overrightarrow{P Q}$ , $\overrightarrow{Q P}$ .

Find the midpoint $M$ of the line segment with endpoints $G(5,4)$ and $H(7,4)$ . Write $M$ as decimals or integers. $M=$

Rotate the vector $(-5,3)$ using the matrix $\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]$ . Find the terminal $x$ -value. $x=$

Rotate the vector $\begin{bmatrix}-5 \\ -3\end{bmatrix}$ using the matrix $\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$ . Find the $x$ -value of the new vector. $x=$

A boat travels at 40 knots on a course of $65^{\circ}$ for 2 hours, then $155^{\circ}$ for 4 hours. Find the distance and bearing from Fort Lauderdale.

Find the vertices of parallelogram $A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} D^{\prime \prime}$ after dilation by 4 and reflection. Are the parallelograms similar or congruent?

Find the midpoint of the segment with endpoints $(5,11)$ and $(3,1)$ .

Find the midpoint of the line segment between the points $(-2,4)$ and $(2,6)$ .

Find the midpoint of the segment between the points $(-7,5)$ and $(7,7)$ .

Sketch an angle $\theta$ with the point $(-6,0)$ on its terminal side. Find the six trig functions of $\theta$ .

Find the new coordinates of B (-5,-8), C (-5,-3), D (0,-3), E (0,-8) after a $180^{\circ}$ rotation around the origin.

Find the new coordinates of vertices B $(2,-9)$ , C $(2,-4)$ , and D $(1,-9)$ after a $270^{\circ}$ clockwise rotation.

Sketch the figure for $\ddot{XY} \perp \overrightarrow{YZ}$ . What’s the first step: A, B, C, or D?

Find point $B$ on line segment $\overline{AC}$ such that the ratio $AB : AC = 1 : 3$ where $A=(-2,4)$ and $C=(4,7)$ .

Find the pre-image of vertex $A^{\prime}(8,5)$ after reflection across the $y$ -axis. Options: $(-8,-6)$ , $(-6,8)$ , $(8,6)$ , $(6,-8)$ .

Find which reflection of the segment from $(-4,-6)$ to $(-6,4)$ gives endpoints $(4,-6)$ and $(6,4)$ .

Find the midpoint of the line segment with endpoints $K(7,7)$ and $L(1,3)$ . Choose from the options: A. $(-3,-2)$ B. $(3,-2)$ C. $(-3,2)$ D. $(3,2)$ .

Find the midpoint of the segment with endpoints $(-2,3)$ and $(5,-3)$ .

Identify the opposite ray to $\overrightarrow{N M}$ from the options: $\overrightarrow{N O}$ , $M$ , $\overrightarrow{N P}$ , $\overrightarrow{M P}$ , or all of the above.

Find the midpoint of the segment with endpoints $(-9,-2)$ and $(-1,-8)$ .

Find the coordinates that divide the line segment from $(2,-6)$ to $(5,6)$ in a 1:5 ratio.

Graph the reflection of the point $T(10,-2)$ over the $x$ -axis.

Reflect the point $S(0,2)$ over the $x$ -axis. What are the coordinates of $S^{\prime}$ ?

Find $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-3 \mathbf{v}$ for $u=\begin{bmatrix}-3 \\ 6\end{bmatrix}, v=\begin{bmatrix}-5 \\ 4\end{bmatrix}$ .

Find $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-3 \mathbf{v}$ for $u=\begin{bmatrix}-3 \\ 6\end{bmatrix}$ and $v=\begin{bmatrix}-5 \\ 4\end{bmatrix}$ .

Is vector $b$ a linear combination of $a_{1}, a_{2}, a_{3}$ ? Choose A, B, C, or D based on the echelon matrix pivots.

Find the vectors $\overrightarrow{P Q}$ and $\overrightarrow{P R}$ for points $P(-2,1)$ , $Q(5,3)$ , and $R(x,y)$ .

Do the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$ span $\mathbb{R}^{4}$ ? Choose A, B, C, or D for the answer.

Find the translation from point $Q(-9,-5)$ to $Q^{\prime}(-2,-8)$ as $\langle x, y \rangle$ .

Find the translation from point $Q(-9,-5)$ to $Q^{\prime}(-2,-8)$ as $\langle x, y\rangle$ . Show your work.

Find the translation from $Q(3,6)$ to $Q^{\prime}(9,3)$ as $\langle x, y\rangle$ . Show your work.

Find the unit vector in the direction of the sum of the vectors $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+\hat{j}+3 \hat{k}$ .

Find the distance, midpoint, and slope of the line between points $(2,-2)$ and $(5,2)$ .

Translate the vertices A(2,-6), B(-1,-1), C(-3,-5) by $(x, y) \rightarrow (x+3, y+5)$ . Find A', B', C'.