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Problem 301

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3. Exkurs: Spurpunkte mit Anwendungen
In diesem Abschnitt werden als exemplarische Anwendungsbeispiele für Geraden Spurpunk, probleme behandell. Die Schnittpunkte einer Geraden mit den Koordinatenebenen bezeichnet man als Spurpunkte der Geraden.
Beispiel: Spurpunkte $\text { Gegeben sei g: } \vec{x}=\left(\begin{array}{l} 2 \\ 4 \\ 2 \end{array}\right)+r\left(\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right)$
Bestimmen Sie die Spurpunkte der Ge raden und fertigen Sie eine Skizze an.
Lösung: Der Schnittpunkt der Geraden mit der $x-y$ Ebene wird als Spurpunkt $S_{x y}$ bezeichnet. Er hat die $z$ -Koordinate $z=0$ . Die $z$ -Koordinate des allgemeinen Ge radenpunktes beträgt $z=2-r$ . Setzen wir diese 0 , so erhalten wir $r=2$ , was auf den Spurpunkt $S_{x y}(4|6| 0)$ führt. $\begin{array}{l} z=0: \Leftrightarrow 2-r=0 \quad \Leftrightarrow r=2 \\ \vec{x}=\left(\begin{array}{l} 2 \\ 4 \\ 2 \end{array}\right)+2 \cdot\left(\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right)=\left(\begin{array}{l} 4 \\ 6 \\ 0 \end{array}\right) \\ S_{x y}(4|6| 0) \end{array}$
Analog errechnen wir die weiteren Spurpunkte, indem wir die x-Koordinate bzw. die y-Koordinate des allgemeinen Geradenpunktes null setzen. - Ergebnisse: $S_{y z}(0|2| 4), S_{x z}(-2|0| 6)$
Übung 1 Berechnen Sie die Spurpunkte der Geraden g durch A und B. Fertigen Sie eine Skizze an. a) $\mathrm{A}(10|6|-1), \mathrm{B}(4|2| 1)$ b) $\mathrm{A}(-2|4| 9), \mathrm{B}(4|-2| 3)$ c) $\mathrm{A}(4|1| 1), \mathrm{B}(-2|1| 7)$ d) $\mathrm{A}(2|4|-2), \mathrm{B}(-1|-2| 4)$
Übung 2 Geben Sie die Gleichung einer Geradeng an, die nur zwei Spurpunkte bzw. nur einen Spurpunkt besitzt.
Übung 3 In welchen Punkten durchdringen die Kanten der skizzierten Pyramide den 2 m hohen Wasserspiegel?

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Problem 302

el vector opuesto de ē: (1,-1,-1) es: a. $\bar{e}=i+j+k$ b. $\bar{e}=(-1,-1,1)$ c. $\bar{e}=\mathrm{i}-\mathrm{j}-\mathrm{k}$ d. $\bar{e}=(-1,-1,1)$

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Problem 303

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 304

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 305

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 306

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

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Problem 307

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

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Problem 308

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 309

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 310

Figure 1 A sled of mass $m$ slides down a rough ramp with a constant speed $v_{0}$ . The angle between the ramp and the horizontal is $\theta$ , as shown in Figure 1. The ramp smoothly transitions to a horizontal surface. The coefficients of static and kinetic friction between the sled and the ramp are $\mu_{s}$ and $\mu_{b}$ respectively. The ramp and the horizontal surface are made of identical materials. (a) The dot in Figure 2 represents the sled when the sled is sliding down the ramp at a constant speed. Draw and label arrows that represent the forces (not components) that are exerted on the sled. Each force in your free-body diagram must be represented by a distinct arrow starting on, and pointing away from the dot.

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Problem 311

What are the coordinates of $R^{\prime}$ for the dilation $D_{(0.5, P)}$ ( $\left.\square P Q R S\right)$ ? ( 3 pts.)
3 $\square$ 4 $\square$ )

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Problem 312

Find the work done by $\mathbf{F}=\left(x^{2}+y\right) \mathbf{i}+\left(y^{2}+x\right) \mathbf{j}+z^{z} \mathbf{k}$ over the following paths from $(2,0,0)$ to $(2,0,4)$ . a. The line segment $x=2, y=0,0 \leq z \leq 4$ b. The helix $r(t)=(2 \cos t) i+(2 \sin t) j+\left(\frac{2 t}{\pi}\right) k, 0 \leq t \leq 2 \pi$ c. The $x$ -axis from $(2,0,0)$ to $(0,0,0)$ followed by the parabola $z=x^{2}, y=0$ from $(0,0,0)$ to $(2,0,4)$
The work done by F over the line segment is $3 e^{\top}+1$ . b. Find $\frac{\mathrm{df}}{\mathrm{dt}}$ for $F$ . A. $\frac{\mathrm{df}}{\mathrm{dt}}=-8 \cos ^{2} \mathrm{t} \sin \mathrm{t}-4 \sin ^{2} \mathrm{t}+\sin ^{2} \mathrm{t} \cos \mathrm{t}+4 \cos ^{2} \mathrm{t}+\frac{4 \mathrm{t}}{\pi^{2}} e^{2 t / \pi}$ B. $\frac{\mathrm{df}}{\mathrm{dt}}=-\sin \mathrm{t}+\cos \mathrm{t}+\frac{1}{\pi} e^{1 / \pi}$ C. $\frac{\mathrm{df}}{\mathrm{dt}}=\frac{1}{3}\left(\cos ^{3} \mathrm{t}\right)+\cos \mathrm{t} \sin \mathrm{t}+\frac{1}{3}\left(\sin ^{3} \mathrm{t}\right)+\frac{2 \mathrm{t}}{\pi} e^{2 \mathrm{t} / \pi}-e^{2 \mathrm{t} / \pi}$ D. $\frac{\mathrm{df}}{\mathrm{dt}}=\cos ^{3} \mathrm{t}+2 \cos \mathrm{t} \sin \mathrm{t}+\sin ^{3} \mathrm{t}+\frac{4 \mathrm{t}}{\pi^{2}} e^{2 \mathrm{t} / \pi}-e^{2 \mathrm{t} / \pi}$

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Problem 313

Find the work done by $F=\left(x^{2}+y\right) i+\left(y^{2}+x\right) j+z e^{z} k$ over the following paths from $(3,0,0)$ to $(3,0,9)$ . a. The line segment $x=3, y=0,0 \leq z \leq 9$ b. The helix $r(t)=(3 \cos t) i+(3 \sin t) j+\left(\frac{9 t}{2 \pi}\right) k, 0 \leq t \leq 2 \pi$ c. The $x$ -axis from $(3,0,0)$ to $(0,0,0)$ followed by the parabola $z=x^{2}, y=0$ from $(0,0,0)$ to $(3,0,9)$ a. Find a scalar potential function $f$ for $F$ , such that $F=\nabla f$ . A. $\frac{1}{3} x^{3}+x^{2} y^{2}+\frac{1}{3} y^{3}+e^{z}+C$ B. $x^{3}+x y+y^{3}+z e^{z}-e^{z}+C$ C. $\frac{1}{3} x^{3}+x y+\frac{1}{3} y^{3}+z-e^{z}+C$ D. $\frac{1}{3} x^{3}+x y+\frac{1}{3} y^{3}+z e^{z}-e^{z}+C$ E. The vector field $\mathbf{F}$ is not conservative.
The work done by $F$ over the line segment is

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Problem 314

Parallelogram $A B C D$ has vertex coordinates $A(0,1), B(1,3), C(4,3)$ , and $D(3$ , 1). It is translated 2 units to the right and 3 units down and then rotated $180^{\circ}$ clockwise around the origin. What are the coordinates of $A$ ? A. $(-2,2)$ B. $(-4,-3)$ C. $(-3,-4)$ D. $(5,2)$

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Problem 315

Un panalelogramo esta conformado por dos vertices conserutiver " a y $b$ ", el vertice $A$ es = $(2,0,0)$ el véntice $b=(0,2,0)$ .'
E centro del paralelogamo is $M=(2,2,2)$ Hallar: a) El perimetro del paralelogramo b) El area de uno de sus trianigulos c) El volumen del panabelepipedo cuya altura es el vector $z=(2,2,4)$

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Problem 316

Compute the curl of the vector field $\mathbf{F}=\langle 5 x-2 y,-(5 y+2 z), 5 z-2 x\rangle$ $\operatorname{curl} \mathbf{F}=$ $\square$

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Problem 317

Calculate the curl of $\mathbf{F}=\left\langle e^{5 y}, \sin ^{7} x, \cos ^{2} x\right\rangle$ . $\operatorname{curl} \mathbf{F}=$ $\square$

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Problem 318

Enercice $A B C$ est un triangle $M$ et $N$ . decux points telo que $\overrightarrow{A M}=\frac{3}{4} \overrightarrow{A B C t} \quad A \vec{N}=\frac{4}{3} \overrightarrow{A C}$ 1) faire la figure 2) on pose $\overrightarrow{B K}=x \overrightarrow{B C}, \overrightarrow{M K}=y \overrightarrow{M N}$ $(x, y) \in \mathbb{R}^{2}$ a-Montrer que $\overrightarrow{M K}=-\frac{3}{4} y \overrightarrow{A B}+\frac{4}{3}$ y $\overrightarrow{A C}$ B- Montrer que $\vec{B} \vec{M}^{2}=\left(\frac{3}{4} y-x\right) \overrightarrow{A B}+\left(x=\frac{4}{3} y\right) A C$ c. Montrer que $\overrightarrow{B M}=-\frac{1}{4} \overrightarrow{A B}$ 3) En considerant, les blifférentes exprescions de Br̈, determiner les valcurs de $x$ et $y$ .

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Problem 319

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 320

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 321

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

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Problem 322

Compute the flux of the vector field $\mathbf{F}=<z, 0,2 x>$ through the surface $S$ with upward orientation and parameterizaton $\mathbf{r}(s, t)=\left\langle s^{2}, 2 s+t^{2}, t\right\rangle, 0 \leq s \leq 1,1 \leq t \leq 2 .$
Flux = $\square$

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Problem 323

1. Each point $A$ is mapped to point $A^{\prime}$ by a dilation centered at the origin with the given scale factor. Complete the table. \begin{tabular}{c|c|c} Coordinates of $\boldsymbol{A}$ & Scale Factor & Coordinates of $\boldsymbol{A}^{\prime}$ \\ \hline $(-4,-2)$ & 3 & \\ \hline $(6,-4)$ & $\frac{1}{2}$ & \\ \hline $(-5,3)$ & 4 & \\ \hline \end{tabular}

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Problem 324

Find the angle $\theta$ between the vectors $\mathbf{v}=2 \mathbf{i}+\mathbf{k}, \mathbf{w}=\mathbf{j}-3 \mathbf{k}$ . $\theta=$ $\square$ degrees
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Problem 325

Find the angle $\theta$ between the vectors $\mathbf{v}=4 \mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{w}=2 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}$ . $\theta=$ $\square$ degrees

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Problem 326

(1 point) Find the following expressions using the graph below of vectors $\mathbf{u}, \mathbf{v}$ , and $\mathbf{w}$ .
1. $\mathbf{u}+\mathbf{v}=$ i+4j
2. $2 u+w=$ $\square$
3. $3 \mathbf{v}-6 \mathbf{w}=$ $\square$
4. $|w|=$ $\square$

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Problem 327

Find the included angle between $\mathbf{u}=\left[\begin{array}{c}0 \\ 0 \\ -1\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0 \\ 1 \\ 1\end{array}\right]$ in $\mathbb{R}^{3}$ . $\theta=$

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Problem 328

Find the volume of the parallelepiped determined by the vectors $\mathbf{u}=\left[\begin{array}{c}1 \\ -1 \\ 1\end{array}\right], \mathbf{v}=$ $\left[\begin{array}{c} -2 \\ -1 \\ 2 \end{array}\right], \text { and } \mathbf{w}=\left[\begin{array}{c} -2 \\ 1 \\ 2 \end{array}\right]$
Volume $=$ $\square$ cubic units

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Problem 329

Pflichtaufgabe 2: (8 Punkte) Gegeben sind die Geraden $g: g: \vec{x}=\left(\begin{array}{c}3 \\ -3 \\ 3\end{array}\right)+r \cdot\left(\begin{array}{c}3 \\ 0 \\ -1\end{array}\right)$ mit $r \in \mathbb{R}$ und $h: \vec{x}=\left(\begin{array}{c}3 \\ -3 \\ 3\end{array}\right)+s \cdot\left(\begin{array}{l}1 \\ 0 \\ 3\end{array}\right)$ mit $s \in \mathbb{R}$ . (1) Geben Sie die Koordinaten des Schnittpunkts von $g$ und $h$ an und zeigen Sie, dass $g$ und $h$ senkrecht zueinander verlaufen. (2) Die Ebene $E$ enthält die Geraden $g$ und $h$ . Prüfen Sie, ob der Punkt $P(7|-3| 5)$ in $E$ liegt.

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Problem 330

40 Trapezold EFGH will be reflected across the $y$ -axis. What will be the resulting coordinate of Point $H^{\prime}$ ? wucan earn 5 coins $(5,-4)$ $(-4,-3)$ $(-5,-2)$ $(5,-2)$

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Problem 331

4. A box weighing 450 N is hanging from two chains attached to an overhead beam at angles $1: 37$ of $70^{\circ}$ and $78^{\circ}$ to the horizontal. a) Draw a vector diagram of this situation. b) Determine the tensions in the chains.

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Problem 332

If $\mathbf{a}=(-1,7,1)$ and $\mathbf{b}=(7,-4,2)$ , find $\mathbf{a} \cdot \mathbf{b}=$

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Problem 333

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 334

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

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Problem 335

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 336

eady Identify Points, Lines, and Rays - Instruction - Level D
Which other figure is a part of line CD? $\overrightarrow{C D}$ $\overrightarrow{D C}$
All of these

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Problem 337

$0 / 1 \mathrm{pts}$
A rigid body is rotating about a fixed axis through the origin. A point on the object located on the $x$ -axis at time $t$ is moving in the positive $z$ direction. What is unit vector in the direction of the angular velocity of the body?

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Problem 338

What is the resultant of $\vec{v} + \vec{u}$ ? Answer in component form. <0, 5> <-4, 5> <-3, 2> <0, -5>

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Problem 339

1. Given the endpoints of a line segment at $A(2,3)$ and $B(8,7)$ , what is the midpoint of A. $(5,5)$ B. $(10,10)$ C. $(6,10)$ D. $(10,5)$ two points Engage Learn

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Problem 340

Evaluate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$ for the given vector field $\mathbf{F}$ and the oriented surface $S$ . In other words, find the flux of $\mathbf{F}$ across $S$ . $\mathbf{F}(x, y, z) = x\mathbf{i} - z\mathbf{j} + y\mathbf{k}$ $S$ is the part of the sphere $x^2 + y^2 + z^2 = 9$ in the first octant, with orientation toward the origin

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Problem 341

Object 1 (mass=7.4 kg) traveling in the +x direction at a speed of 13.84m/s undergoes an inelastic collision with Object 2 (mass=6.10kg) which is at rest. Object two flies off at a speed of 9.95m/s and an angle $\theta_2 = 40.7^\circ$ . The diagram below is just a sketch and is not meant to have accurate angles. It is possible that Object 1 travels back and to the left, though it is not possible for it to travel in the -y direction. (Do you know why?)
What is the total momentum in x direction after the collision?
What is the momentum of Object 1 in the y direction after the collision?
What is the speed of Object 1 after the collision?
What is the angle $\theta_1$ in degrees? (Do not enter units)

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Problem 342

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

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Problem 343

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 344

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 345

Question 6 Find the midpoint of the line segment with endpoints: $\left(-\frac{5}{8},-\frac{7}{5}\right)$ to $\left(\frac{5}{4},-\frac{6}{5}\right)$
Midpoint $=$ $\square$ Give your answer as a point, using integers or reduced fractions for coordinates.
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Problem 346

4. Determine the dot products of the following pairs of column vectors.
(c) $\left[\begin{array}{r}2 \\ 0 \\ -5\end{array}\right],\left[\begin{array}{r}3 \\ 6 \\ -4\end{array}\right]$ $\square$ $\square$ $\square$

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Problem 347

18. If possible, determine the value(s) of $k$ in each case so that the line $[x, y, z]=[k,-4,-6]+t[3,2,1]$ and the plane $x-4 y+5 z=-5$ intersect at the given number of points. a) a single point b) an infinite number of points c) no points
19. Consider these lines. $\ell_{1}:[x, y, z]=[2,1,-4]+t[1,3,1]$

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Problem 348

\begin{array}{l} \text{2.6 Given the semantic feature space:} \\ \text{2.6.1 Calculate} \\ \text{[king] - [prince] + [girl] =} \\ \text{[king] - [prince] + [girl] =} \\ -9 + 9.5 = 8.5 \\ \text{2.6.2 Clarify your answer showing the steps of your calculation.} \end{array}

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Problem 349

```latex \text{2.6 Given the following 30 semantic feature [1+3 Marks]}
\text{2.6.1 Calculate } [\text{king}]-[\text{prince}]+[\text{girl}]=
\text{2.6.2 Clarify your answer showing the steps of your calculation.}
\text{30 Semantic Feature Space} ```

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Problem 350

Instructions: Do 100 points' worth of the following considerations to the best of your ability. Be sure to show your work, as no credit will be given for an unjustified answer. You may use $g = 10.000000 \, m/s^2$ for freefall acceleration due to Earth's gravity. You may also need to know that there are 5280 feet in one mile, 60 seconds in a minute, 60 minutes in an hour, 13 muffins in a baker's dozen, and several geese in a gaggle. You should solve all problems algebraically before plugging in any numbers. Also, apply as many checks as you can to your results! You are permitted one sheet of notes.
1. While saving the world one weekend, The Courageous Dr. Bob must tackle a 140 kg bad guy. Before the collision, the bad guy is running north at $3.0 \, m/s$ , and The 70. kg Courageous Dr. Bob is running west at $4.0 \, m/s$ .
A. If the collision is completely inelastic, what will be the velocity of The Courageous Dr. Bob and the villain after the collision? (15 points) B. What is the change in kinetic energy? (15 points) C. Where did the "missing" energy go? (5 points) D. Estimate the weight of the bad guy in units of pounds. (5 points) E. For part (E) you no-doubt had to find the square of a two-dimensional vector (the final velocity). Perhaps you did that using the Pythagorean Theorem. Show that the Pythagorean Theorem and simply taking the dot product of the final velocity with itself yield the same result. (10 points)

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Problem 351

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 352

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 353

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 354

Let $v = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and let $x = \begin{bmatrix} -1 \\ 8 \end{bmatrix}$ .
The $Proj_v x$ is equal to $k \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ where
k = \text{______}.

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Problem 355

The rocket sled shown in Figure 4.33 accelerates at a rate of $49.0 \text{ m/s}^2$ . Its passenger has a mass of $75.0 \text{ kg}$
Figure 4.33
(a) Calculate the horizontal component of the force the seat exerts against his body.
(In case you are curious) Compare this with his weight by using a ratio.
b) Calculate the direction and magnitude of the total force the seat exerts against his body.
Direction (angle above the horizontal, to the nearest degree)

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Problem 356

A pilot flies in a straight path for 1 hour and 30 min . She then makes a course correction, heading 10 degrees to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?
Answer: $\square$ miles

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Problem 357

Чему равен вектор $\vec{a}$ Выберите один ответ: a. $(-1 ; 2)$ b. $(4 ; 2)$ c. $(-1 ;-2)$ d. $(-1 ; 4)$ e. $(1 ; 2)$

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Problem 358

$A(8, 5, -6)$ , $B(4, 7, 9)$ , $C(2, 1, 6)$ ج) إذا كانت : فبرهن باستخدام المتجهات أن النقط $A$ , $B$ , $C$ تمثل رؤوس مثلث الزاوية

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Problem 359

(d) Show that, if $\vec{a}$ and $\vec{b}$ are non-zero vectors belonging to $\mathbf{R}^{3}$ , then $\vec{a}$ and $\vec{b}$ can't be both orthogonal and parallel

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Problem 360

4. (10 pts) A force of $\mathbf{F}=\left\langle 2,3, \frac{1}{2}\right\rangle$ is used to push an object up a hill from point $A$ to point (shown below). How much work is done? Assume the units of force are Newtons and the units distance are meters.

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Problem 361

Find the surface integral of the field $F(x,y,z) = 4y\mathbf{i} - 4x\mathbf{j} + \mathbf{k}$ across the portion of the sphere $x^2 + y^2 + z^2 = a^2$ in the first octant in the direction away from the origin.
The value of the surface integral is ▢. (Type an exact answer, using $\pi$ as needed.)

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Problem 362

Let $S$ be the portion of the cylinder $y = e^x$ in the first octant that projects parallel to the x-axis onto the rectangle $R_{yz}$ : $1 \le y \le 4$ , $0 \le z \le 3$ in the yz-plane (see the accompanying figure). Let $\mathbf{n}$ be the unit vector normal to $S$ that points away from the yz-plane. Find the flux of the field $\mathbf{F}(x,y,z) = \mathbf{i} - 3y\mathbf{j} + 2z\mathbf{k}$ across $S$ in the direction of $\mathbf{n}$ .
The flux is $\boxed{\phantom{00}}$ . (Type an exact answer.)

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Problem 363

Let $S$ be the cylinder $x^2 + y^2 = a^2$ , $0 \le z \le h$ , together with its top, $x^2 + y^2 \le a^2$ , $z = h$ . Let $\mathbf{F} = -2y\mathbf{i} + 2x\mathbf{j} + 2x^2\mathbf{k}$ . Use Stokes' Theorem to find the flux of $\nabla \times \mathbf{F}$ through $S$ in the direction away from the interior of the cylinder.
The flux of $\nabla \times \mathbf{F}$ outward through $S$ is $\boxed{\phantom{0}}$ . (Type an exact answer, using $\pi$ as needed.)

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Problem 364

Find the midpoint of the segment with the following endpoints. $(0, 8)$ and $(6, 4)$

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Problem 365

The vectors $\mathbf{u}$ and $\mathbf{v}$ have the same direction. a. Find $\|\mathbf{u}\|$ . b. Find $\|v\|$ . c. Is $\mathbf{u}=\mathbf{v}$ ? Explain. a. $\|\mathbf{u}\|=$ $\square$ (Simplify your answer. Type an exact answer, using radicals as needed.)

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Problem 366

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 367

Use curl to determine whether the vector field $\mathbf{F} = \langle 3x^6 y, x^7 + 3y^8 z, y^9 \rangle$ is conservative.
(a) Find curl $\mathbf{F}$ .
curl $\mathbf{F}$ = $\langle \hphantom{a}, \hphantom{a}, \hphantom{a} \rangle$ .
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Problem 368

Identify the transformation based on the coordinates below. Figure 1: $E$ (-9, 3), $F$ (-8, 5), $G$ (-5, 5), $H$ (-4, 3) Figure 1': $E'$ (9, -3), $F'$ (8, -5), $G'$ (5, -5), $H'$ (4, -3) $90^\circ$ rotation (clockwise) $180^\circ$ rotation dilation

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Problem 369

Triangle $ABC$ will be rotated $90^\circ$ counter-clockwise about the origin. What will be the resulting coordinate of Point $A'$ ?

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Problem 370

1. The three cables (AD, AC and AB) are used to support the 40 kg flowerpot. Determine the force developed in cable AD.
a) 2220 N b) 763 N c) 262 N d) 77.7 N e) None of the above

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Problem 371

Knowledge Clieck Question 3
A dilation centered at the origin with a scale factor of $\frac{2}{3}$ is applied to $\triangle X Y Z$ . The result is $\triangle X^{\prime} Y^{\prime} Z^{\prime}$ , as shown below. (a) The arrows below show that the coordinates on the left are mapped to the coordinates on the right. Fill in the blanks to glve the coordinates after the dilation. original coordinates $-\boldsymbol{f i n a l}$ coordinates $\left.\begin{array}{r} X(3,6) \rightarrow X^{\prime}(\square, \square) \\ Y(6,-9) \\ Z\left(-Y^{\prime}(\square, \square)\right. \\ Z(-12,-3) \end{array}\right)$ (b) Choose the general rule below that describes the dilation mapping $\triangle X Y Z$ to $\triangle X^{\prime} Y^{\prime} Z^{\prime}$ . $(x, y) \rightarrow\left(\frac{3}{2} y, \frac{3}{2} x\right)$ $(x, y) \rightarrow\left(\frac{2}{3} y, \frac{2}{3} x\right)$ $(x, y) \rightarrow\left(\frac{2}{3} x, \frac{3}{2} y\right)$ $(x, y) \rightarrow\left(\frac{3}{2} x, \frac{2}{3} y\right)$ $(x, y) \rightarrow\left(\frac{3}{2} x, \frac{3}{2} y\right)$ $(x, y) \rightarrow\left(x, \frac{2}{3} y\right)$ $(x, y) \rightarrow\left(\frac{2}{3} x, y\right)$ $(x, y) \rightarrow\left(\frac{2}{3} x, \frac{2}{3} y\right)$ I Don't Know Submit O 2024 McGraw Hill LLC. All Righ

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Problem 372

Top view of a $3.0 \text{ cm} \times 3.0 \text{ cm} \times 3.0 \text{ cm}$ cube
$4.0 \text{ m}$ $400 \text{ N/C}$ $2.0 \text{ m}$ $30^\circ$ $500 \text{ N/C}$ $30^\circ$ 39. 40.
FIGURE P24.29 FIGURE P24.30
30. FIGURE P24.30 shows four sides of a $3.0 \text{ cm} \times 3.0 \text{ cm} \times 3.0 \text{ cm}$ cube.
a. What are the electric fluxes $\Phi_1$ to $\Phi_4$ through sides 1 to 4? b. What is the net flux through these four sides?

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Problem 373

In another experiment on the same table, the target ball $B$ is replaced by target ball $C$ of mass 0.10 kg . The incident ball $A$ again slides at $1.4 \mathrm{~m} / \mathrm{s}$ , as shown above left, but this time makes a glancing collision with the target ball $C$ that is at rest at the edge of the table. The target ball $C$ strikes the floor at point $P$ , which is at a horizontal displacement of 0.15 m from the point of the collision, and at a horizontal angle of $30^{\circ}$ from the $+x$ -axis, as shown above right. c. Calculate the speed vof the target ball Cimmediately after the collision. d. Calculate the $y$ -component of incident ball $A^{\prime}$ 's momentum immediately after the collision.
Note On your AP Exam, you will handwrite your responses to free-response questions in a test

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Problem 374

a. A landscaper pushes a lawnmower forward on flat ground with a displacement of 1000 ft with a force vector $\overrightarrow{\mathrm{F}}=\langle 26,-15\rangle$ , where the force is in pounds. How much work is done by the landscaper? b. Now, analyze the situation in part (a). If a lawnmower is being pushed by its handle, explain the components of the force vector.

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Problem 375

Question Show Examples
The figure below is rotated $180^{\circ}$ clockwise and then reflected over y -axis. What are the coordinates of the image of point V after these transformations? Answer Astempt 2 out of 2

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Problem 376

If $\vec{u}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{v}=3 \hat{i}-\hat{j}+3 \hat{k}$ then $(\vec{u} \times \vec{v}) \cdot(\vec{u}-\vec{v})+\vec{u} \cdot \vec{v}$ equals

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Problem 377

C) Dati i punti $A(-2;1)$ , $B(1;-1)$ , $D(2;7)$ , e la retta $r$ di equazione $2x-y-7=0$ 1)Si determini la misura dell'angolo DAB (1 punto) 2)Si determinino le coordinate del punto $C$ appartenente a $r$ tale che i segmenti $BC$ e $AD$ siano paralleli (1 punto) 3)Si determini l'area del quadrilatero $ABCD$ (1 punto)

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Problem 378

Represent the vector $\mathbf{v}$ in the form $\mathbf{v}=\mathrm{a} \mathbf{i}+\mathrm{b} j$ $\|\mathbf{v}\|=34 ; \theta=225^{\circ}$

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Problem 379

6. Grundwissen:
Eine Vogelfeder schwebt mit konstanter Geschwindigkeit nach unten. Welche Kräfte wirken hier auf die Feder? Kurze Begründung.
Ich wünsche dir viel Erfolg und eine frohe Weihnachtszeit! Gr

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Problem 380

Exercice 2 ABC un triangle quelconque avec $\mathrm{AB}=\mathrm{c} ; \mathrm{AC}=\mathrm{b} ; \mathrm{BC}=\mathrm{a}$ . et $\mathrm{A}^{\prime}$ et $\mathrm{B}^{\prime}$ sont les, pieds respectifs des bissectrices intérieures issues de $A$ et $B$ qui se coupe en I centré du cercle inscrits et la parallçle à (AD) en B coupe (AC) en D. 1) Montrer que $A B D$ est isocelle en $A$ et en déduire que $\frac{A^{\prime} B}{A^{\prime} C}=\frac{A D}{A C}=\frac{c}{a}$ et que $\mathrm{A}^{\prime}=\operatorname{bar}\{(\mathrm{B}, \mathrm{b}) ;(\mathrm{C}, \mathrm{c})\}$ puis exprimer de même $\mathrm{B}^{\prime}$ comme barycentre de A et C 2) En déduire que : $1=\operatorname{bar}\{(A, a) ;(B, b) ;(C, c)\}$ 3) Soit $H$ lorthocentre de $A B C$ , montrer que : $\vec{H}=$ bar $\{(A, \operatorname{tg} \widehat{A}) ;(B, \operatorname{tg} \widehat{B}) ;(C, \operatorname{tg} \widehat{C})\}$

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Problem 381

The point B (-2, 1) has been transformed to B' (-5, -3). The transformation is described as _____.
$T_{(-3,-4)}$ $T_{(-3,-2)}$ $R_{x=2}$ $D_3$

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Problem 382

The vector $i$ represents 1 mile per hour east, and the vector $j$ represents 1 mile per hour north. Maria is jogging south at 12 miles per hour. One of the following vectors represents Maria's velocity, in miles per hour. Which one?
A. $-12i$ B. $-12j$ C. $12i$ D. $12j$ E. $12i + 12j$

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Problem 383

The magnitude of a vector $\mathbf{t}$ is 5 and its direction angle $\theta$ is $180^{\circ}$ . Write the component form for $t$ .
Write each component in exact simplified form.
Save answer

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Problem 384

(a) [2 pt] Let $\vec{u} = 4\vec{i} + 2\vec{j}$ and let $\vec{v} = -6\vec{i} + \vec{j}$ . Compute the following sum. $\vec{u} + \vec{v} =$ $-2\vec{i} + 3\vec{j}$ (b) [2 pt] Let $\vec{u} = (3, -9)$ . Compute the following scalar product. $-5\vec{u} =$ (c) [2 pt] Let $\vec{u} = (8, 7, 10, 5)$ and $\vec{v} = (8, 6, 9, 7)$ . Compute the following vector. $\frac{\vec{u} + \vec{v}}{2} =$ (d) Let $\vec{u} = (2, -2, 1)$ , and let $\vec{v} = (-2, 6, 2)$ . i. [1 pt] Compute the magnitude of $\vec{u}$ . $||\vec{u}|| =$ ii. [1 pt] Compute the magnitude of $\vec{v}$ . $||\vec{v}|| =$ iii. [2 pt] Compute the magnitude of $\vec{u} + \vec{v}$ . $||\vec{u} + \vec{v}|| =$

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Problem 385

Find the distance between the pair of points. 7) $(-3, -1)$ and $(-11, 5)$

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Problem 386

7. Find $t$ given that these vectors are perpendicular: a. $\mathbf{p} = \begin{pmatrix} 3 \\ t \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} -2 \\ 1 \end{pmatrix}$ b. $\mathbf{r} = \begin{pmatrix} t \\ t+2 \end{pmatrix}$ and $\mathbf{s} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ c. $\mathbf{a} = \begin{pmatrix} t \\ t+2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2-3t \\ t \end{pmatrix}$ d. $\mathbf{a} = \begin{pmatrix} 3 \\ -1 \\ t \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2t \\ -3 \\ -4 \end{pmatrix}$
8. For question 7 find, where possible, the value(s) of $t$ for which the given vectors are

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Problem 387

Find the flux of the field $F(x,y,z) = z^3\mathbf{i} + x\mathbf{j} - 6z\mathbf{k}$ outward through the surface cut from the parabolic cylinder $z = 4 - y^2$ by the planes $x = 0$ , $x = 1$ , and $z = 0$ .
The flux is $\boxed{}$ . (Simplify your answer.)

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Problem 388

A body of weight $P = 8$ N, considered as a material point, slides on an inclined plane of length $AB = L = 0.5$ m with no initial velocity. The contact is characterized by a coefficient of kinetic friction $\mu_k = 0.40$ . Take $g = 10 \text{m}\cdot\text{s}^{-2}$ .
1. Represent the forces acting on the body.
2. Find the forces expressions and calculate their magnitudes.
3. Calculate the acceleration $a$ of the body.
4. Calculate the velocity of the body at point B. Grading scale in order: $1+2+1+1=5$ *Numbering the answers and improving the handwriting is necessary.

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Problem 389

Find the area of the triangle defined by the vectors $\langle 4,8\rangle$ and $\langle-1,2\rangle$ . $\square$ square units (Type an integer or a fraction.)

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Problem 390

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 391

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 392

5) A $860-\mathrm{kg}$ Escalade traveling $40 \mathrm{~m} / \mathrm{s}$ @ 120 degrees has a perfect inelastic collision with a 220 -kg mini-cooper traveling $22 \mathrm{~m} / \mathrm{s} @ 73$ degrees. Find with what velocity the two cars continue moving after the collision.

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Problem 393

Exercise 4.1.17 In each case, find the point $Q$ : a. $\overrightarrow{P Q}=\left[\begin{array}{r}2 \\ 0 \\ -3\end{array}\right]$ and $P=P(2,-3,1)$

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Problem 394

1. Consider the parallelogram given by the coordinates $P=(0,0,4), Q=(2,0,0)$ , $R=(0,3,0)$ , and a fourth unknown coordinate $S$ , and additionaly we have $Q R$ parallel to PS min (a) Determine the location of the point $S$ . (b) Calculate the area of the parallelogram $\mathcal{P}=P Q R S$ . (c) Determine the equation of the plane containing the parallelogram. Please put you answer in the form of $a x+b y+c z=d$ .

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Problem 395

A line segment has endpoints at $(-4,-6)$ and $(-6,4)$ . Which reflection will produce an image with endpoints at $(4,-$ $6)$ and $(6,4)$ ?
a reflection of the line segment across the $x$ -axis a reflection of the line segment across the $y$ -axis a reflection of the line segment across the line $y=x$ a reflection of the line segment across the line $y=-x$

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Problem 396

6. You are pushing a rock along level ground and making the rock speed up. Illustrate the simultaneous forces between two objects, as represented in Newton's third law of motion, to identify how the size of the force you exert on the rock compares with the size of the force the rock exerts on you. The force you exert $\qquad$

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Problem 397

Construct free-body diagrams for the following objects; label the forces according to type. Use the approximation that $g=-10 \mathrm{~m} / \mathrm{s}^{2}$ to determine the magnitude of all forces and the net force and acceleration of the object.
1. A 2-kg box is free-falling from the table to the ground. $F_{\text {gravity }}=m_{1}=2 \mathrm{~kg} \cdot 1 \mathrm{~m} / \mathrm{s}^{2}$ Fgav: 1 y = 20 N Fot $=20 \mathrm{~N}$ $a=\frac{F_{m o t}}{m}=\frac{20 N}{2 k y}=10 y_{s}{ }^{2}$
4. A 500-kg freight elevator is descending down through the shaft at a constant velocity of $0.50 \mathrm{~m} / \mathrm{s}$ .
2. An 8-N force is applied to a 2-kg box to move it to the right across the table at a constant velocity of $1.5 \mathrm{~m} / \mathrm{s}$ .
5. A 500-kg freight elevator is moving upwards towards its destination. Near the end of the ascent, the upward moving elevator encounters a downward acceleration of $2.0 \mathrm{~m} / \mathrm{s}^{2}$ .
3. An 8 -N force is applied to a $2-\mathrm{kg}$ box to accelerate it to the right across a table. The box encounters a force of friction of 5 N .

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Problem 398

What is the angle between the vectors $\mathbf{u}=\left[\begin{array}{l}2 \\ 0 \\ 0 \\ 0\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right]$ ? a) 0 b) $\pi / 6$ c) $\pi / 4$ d) $\pi / 3$ e) $\pi / 2$

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Problem 399

Find the midpoint of the line segment joining the points $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ . $P_{1}=(3,-3) ; P_{2}=(5,3)$
The midpoint of the line segment joining the points $P_{1}$ and $P_{2}$ is $\square$ . (Simplify your answer. Type an ordered pair.)

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Problem 400

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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Vector

Problem 301

Problem 302

el vector opuesto de ē: (1,-1,-1) es: a. eˉ=i+j+k\bar{e}=i+j+keˉ=i+j+k b. eˉ=(−1,−1,1)\bar{e}=(-1,-1,1)eˉ=(−1,−1,1) c. eˉ=i−j−k\bar{e}=\mathrm{i}-\mathrm{j}-\mathrm{k}eˉ=i−j−k d. eˉ=(−1,−1,1)\bar{e}=(-1,-1,1)eˉ=(−1,−1,1)

Problem 303

Problem 304

Do the vectors v1,v2,v3\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}v1​,v2​,v3​ span R4\mathbb{R}^{4}R4? Choose A, B, C, or D for the answer.

Problem 305

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).

Problem 306

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

Problem 307

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

Problem 308

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

Problem 309

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

Problem 310

Problem 311

What are the coordinates of R′R^{\prime}R′ for the dilation D(0.5,P)D_{(0.5, P)}D(0.5,P)​ ( □PQRS)\left.\square P Q R S\right)□PQRS) ? ( 3 pts.)3 □\square□ 4 □\square□ )

Problem 312

Problem 313

Problem 314

Problem 315

Problem 316

Compute the curl of the vector field F=⟨5x−2y,−(5y+2z),5z−2x⟩\mathbf{F}=\langle 5 x-2 y,-(5 y+2 z), 5 z-2 x\rangleF=⟨5x−2y,−(5y+2z),5z−2x⟩ curl⁡F=\operatorname{curl} \mathbf{F}=curlF= □\square□

Problem 317

Calculate the curl of F=⟨e5y,sin⁡7x,cos⁡2x⟩\mathbf{F}=\left\langle e^{5 y}, \sin ^{7} x, \cos ^{2} x\right\rangleF=⟨e5y,sin7x,cos2x⟩. curl⁡F=\operatorname{curl} \mathbf{F}=curlF= □\square□

Problem 318

Problem 319

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.

Problem 320

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

Problem 321

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

Problem 322

Problem 323

Problem 324

Find the angle θ\thetaθ between the vectors v=2i+k,w=j−3k\mathbf{v}=2 \mathbf{i}+\mathbf{k}, \mathbf{w}=\mathbf{j}-3 \mathbf{k}v=2i+k,w=j−3k. θ=\theta=θ= □\square□ degreesPreview My Answers Submit Answers Your score was recorded. You have attempted this problem 1 time.

Problem 325

Find the angle θ\thetaθ between the vectors v=4i−j+k,w=2i+3j+5k\mathbf{v}=4 \mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{w}=2 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}v=4i−j+k,w=2i+3j+5k. θ=\theta=θ= □\square□ degrees

Problem 326

(1 point) Find the following expressions using the graph below of vectors u,v\mathbf{u}, \mathbf{v}u,v, and w\mathbf{w}w.1. u+v=\mathbf{u}+\mathbf{v}=u+v= i+4j2. 2u+w=2 u+w=2u+w= □\square□3. 3v−6w=3 \mathbf{v}-6 \mathbf{w}=3v−6w= □\square□4. ∣w∣=|w|=∣w∣= □\square□

Problem 327

Find the included angle between u=[00−1]\mathbf{u}=\left[\begin{array}{c}0 \\ 0 \\ -1\end{array}\right]u=⎣⎡​00−1​⎦⎤​ and v=[011]\mathbf{v}=\left[\begin{array}{c}0 \\ 1 \\ 1\end{array}\right]v=⎣⎡​011​⎦⎤​ in R3\mathbb{R}^{3}R3. θ=\theta=θ=

Problem 328

Problem 329

Problem 330

40 Trapezold EFGH will be reflected across the yyy-axis. What will be the resulting coordinate of Point H′H^{\prime}H′ ? wucan earn 5 coins (5,−4)(5,-4)(5,−4) (−4,−3)(-4,-3)(−4,−3) (−5,−2)(-5,-2)(−5,−2) (5,−2)(5,-2)(5,−2)

Problem 331

4. A box weighing 450 N is hanging from two chains attached to an overhead beam at angles 1:371: 371:37 of 70∘70^{\circ}70∘ and 78∘78^{\circ}78∘ to the horizontal. a) Draw a vector diagram of this situation. b) Determine the tensions in the chains.

Problem 332

If a=(−1,7,1)\mathbf{a}=(-1,7,1)a=(−1,7,1) and b=(7,−4,2)\mathbf{b}=(7,-4,2)b=(7,−4,2), find a⋅b=\mathbf{a} \cdot \mathbf{b}=a⋅b=

Problem 333

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

Problem 334

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

Problem 335

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

Problem 336

eady Identify Points, Lines, and Rays - Instruction - Level DWhich other figure is a part of line CD? CDundefined\overrightarrow{C D}CD DCundefined\overrightarrow{D C}DCAll of these

Problem 337

0/1pts0 / 1 \mathrm{pts}0/1ptsA rigid body is rotating about a fixed axis through the origin. A point on the object located on the xxx-axis at time ttt is moving in the positive zzz direction. What is unit vector in the direction of the angular velocity of the body?

Problem 338

What is the resultant of v⃗+u⃗\vec{v} + \vec{u}v+u? Answer in component form. <0, 5> <-4, 5> <-3, 2> <0, -5>

Problem 339

1. Given the endpoints of a line segment at A(2,3)A(2,3)A(2,3) and B(8,7)B(8,7)B(8,7), what is the midpoint of A. (5,5)(5,5)(5,5) B. (10,10)(10,10)(10,10) C. (6,10)(6,10)(6,10) D. (10,5)(10,5)(10,5) two points Engage Learn

Problem 340

Problem 341

Problem 342

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

Problem 343

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

Problem 344

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

Problem 345

Problem 346

4. Determine the dot products of the following pairs of column vectors.(c) [20−5],[36−4]\left[\begin{array}{r}2 \\ 0 \\ -5\end{array}\right],\left[\begin{array}{r}3 \\ 6 \\ -4\end{array}\right]⎣⎡​20−5​⎦⎤​,⎣⎡​36−4​⎦⎤​ □\square□ □\square□ □\square□

Problem 347

Problem 348

el vector opuesto de ē: (1,-1,-1) es: a. $\bar{e}=i+j+k$ b. $\bar{e}=(-1,-1,1)$ c. $\bar{e}=\mathrm{i}-\mathrm{j}-\mathrm{k}$ d. $\bar{e}=(-1,-1,1)$

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.

What are the coordinates of $R^{\prime}$ for the dilation $D_{(0.5, P)}$ ( $\left.\square P Q R S\right)$ ? ( 3 pts.)
3 $\square$ 4 $\square$ )

Compute the curl of the vector field $\mathbf{F}=\langle 5 x-2 y,-(5 y+2 z), 5 z-2 x\rangle$ $\operatorname{curl} \mathbf{F}=$ $\square$

Calculate the curl of $\mathbf{F}=\left\langle e^{5 y}, \sin ^{7} x, \cos ^{2} x\right\rangle$ . $\operatorname{curl} \mathbf{F}=$ $\square$

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 $(5,2)$ .

Find the angle $\theta$ between the vectors $\mathbf{v}=2 \mathbf{i}+\mathbf{k}, \mathbf{w}=\mathbf{j}-3 \mathbf{k}$ . $\theta=$ $\square$ degrees
Preview My Answers Submit Answers Your score was recorded. You have attempted this problem 1 time.

Find the angle $\theta$ between the vectors $\mathbf{v}=4 \mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{w}=2 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}$ . $\theta=$ $\square$ degrees

(1 point) Find the following expressions using the graph below of vectors $\mathbf{u}, \mathbf{v}$ , and $\mathbf{w}$ .
1. $\mathbf{u}+\mathbf{v}=$ i+4j
2. $2 u+w=$ $\square$
3. $3 \mathbf{v}-6 \mathbf{w}=$ $\square$
4. $|w|=$ $\square$

Find the included angle between $\mathbf{u}=\left[\begin{array}{c}0 \\ 0 \\ -1\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}0 \\ 1 \\ 1\end{array}\right]$ in $\mathbb{R}^{3}$ . $\theta=$

40 Trapezold EFGH will be reflected across the $y$ -axis. What will be the resulting coordinate of Point $H^{\prime}$ ? wucan earn 5 coins $(5,-4)$ $(-4,-3)$ $(-5,-2)$ $(5,-2)$

4. A box weighing 450 N is hanging from two chains attached to an overhead beam at angles $1: 37$ of $70^{\circ}$ and $78^{\circ}$ to the horizontal. a) Draw a vector diagram of this situation. b) Determine the tensions in the chains.

If $\mathbf{a}=(-1,7,1)$ and $\mathbf{b}=(7,-4,2)$ , find $\mathbf{a} \cdot \mathbf{b}=$

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

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

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

eady Identify Points, Lines, and Rays - Instruction - Level D
Which other figure is a part of line CD? $\overrightarrow{C D}$ $\overrightarrow{D C}$
All of these

$0 / 1 \mathrm{pts}$
A rigid body is rotating about a fixed axis through the origin. A point on the object located on the $x$ -axis at time $t$ is moving in the positive $z$ direction. What is unit vector in the direction of the angular velocity of the body?

What is the resultant of $\vec{v} + \vec{u}$ ? Answer in component form. <0, 5> <-4, 5> <-3, 2> <0, -5>

1. Given the endpoints of a line segment at $A(2,3)$ and $B(8,7)$ , what is the midpoint of A. $(5,5)$ B. $(10,10)$ C. $(6,10)$ D. $(10,5)$ two points Engage Learn

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

4. Determine the dot products of the following pairs of column vectors.
(c) $\left[\begin{array}{r}2 \\ 0 \\ -5\end{array}\right],\left[\begin{array}{r}3 \\ 6 \\ -4\end{array}\right]$ $\square$ $\square$ $\square$

```latex \text{2.6 Given the following 30 semantic feature [1+3 Marks]}
\text{2.6.1 Calculate } [\text{king}]-[\text{prince}]+[\text{girl}]=
\text{2.6.2 Clarify your answer showing the steps of your calculation.}
\text{30 Semantic Feature Space} ```

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

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. $T_{(4,6)}$ .

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

Let $v = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and let $x = \begin{bmatrix} -1 \\ 8 \end{bmatrix}$ .
The $Proj_v x$ is equal to $k \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ where
k = \text{______}.

Чему равен вектор $\vec{a}$ Выберите один ответ: a. $(-1 ; 2)$ b. $(4 ; 2)$ c. $(-1 ;-2)$ d. $(-1 ; 4)$ e. $(1 ; 2)$

$A(8, 5, -6)$ , $B(4, 7, 9)$ , $C(2, 1, 6)$ ج) إذا كانت : فبرهن باستخدام المتجهات أن النقط $A$ , $B$ , $C$ تمثل رؤوس مثلث الزاوية

(d) Show that, if $\vec{a}$ and $\vec{b}$ are non-zero vectors belonging to $\mathbf{R}^{3}$ , then $\vec{a}$ and $\vec{b}$ can't be both orthogonal and parallel

4. (10 pts) A force of $\mathbf{F}=\left\langle 2,3, \frac{1}{2}\right\rangle$ is used to push an object up a hill from point $A$ to point (shown below). How much work is done? Assume the units of force are Newtons and the units distance are meters.

Find the midpoint of the segment with the following endpoints. $(0, 8)$ and $(6, 4)$

The vectors $\mathbf{u}$ and $\mathbf{v}$ have the same direction. a. Find $\|\mathbf{u}\|$ . b. Find $\|v\|$ . c. Is $\mathbf{u}=\mathbf{v}$ ? Explain. a. $\|\mathbf{u}\|=$ $\square$ (Simplify your answer. Type an exact answer, using radicals as needed.)

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].

Identify the transformation based on the coordinates below. Figure 1: $E$ (-9, 3), $F$ (-8, 5), $G$ (-5, 5), $H$ (-4, 3) Figure 1': $E'$ (9, -3), $F'$ (8, -5), $G'$ (5, -5), $H'$ (4, -3) $90^\circ$ rotation (clockwise) $180^\circ$ rotation dilation

Triangle $ABC$ will be rotated $90^\circ$ counter-clockwise about the origin. What will be the resulting coordinate of Point $A'$ ?

1. The three cables (AD, AC and AB) are used to support the 40 kg flowerpot. Determine the force developed in cable AD.
a) 2220 N b) 763 N c) 262 N d) 77.7 N e) None of the above