Vector

Problem 401

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 402

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 403

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 404

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 405

A point charge of $2 \mu \mathrm{C}$ is placed at the origin. There is an external uniform field $\mathbf{E}=600 \mathrm{iN} / \mathrm{C}$ . What is the net force on a $6 \mu \mathrm{C}$ charge placed at $(3 \mathrm{~m}, 3 \mathrm{~m})$ ? $7.843 \cdot 10^{-3} \mathrm{j}+4.243 \cdot 10^{-3} \mathrm{jN}$
Note: You can earn partial credit on this problem. Show: $\square$ Correct Answers

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Problem 406

$\text{جد قيم } \, x \, \text{التي تجعل الزاوية بين المتجهين منفرجة}$

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Problem 407

Sketch the polar point (-6, -25π/12).

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Problem 408

Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^{4}$ as that spanned by the vectors $\left[\begin{array}{l} 2 \\ 2 \\ 0 \\ 1 \end{array}\right], \quad\left[\begin{array}{c} -4 \\ -2 \\ 2 \\ -5 \end{array}\right], \quad\left[\begin{array}{c} 3 \\ 2 \\ -1 \\ 3 \end{array}\right], \quad\left[\begin{array}{c} 7 \\ 6 \\ -1 \\ 5 \end{array}\right]$
A linearly independent spanning set for the subspace is: $\left\{\begin{array}{l} {\left[\begin{array}{l} \square \\ \square \\ \square \\ \square \end{array}\right],\left[\begin{array}{l} {\left[\begin{array}{l} \square \\ \square \\ \square \\ \square \end{array}\right]} \end{array}\right\} . . . . . . . . . . . ~} \\ \square \end{array}\right]$

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Problem 409

\text{How do you know that the bicyclists in the first panel above are moving at a constant velocity?}

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Problem 410

Relevante Lernziele: Lineare Algebra Gegeben ist ein Dreieck mit den Eckpunkten $A(5,3,1), B(1,1,4)$ und $C(4,5,4)$ . Bestimmen Sie die Längen der Seiten und die Innenwinkel des Dreiecks.

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Problem 411

Question 10 : A rectangular metal frame LMOK is placed between two long and parallel straight wires, all of which are in the same plane as the figure, if the same current intensity I passes through each of them, then the frame ..... 1- rotates around an axis parallel to the two wires 2- is not affected by a torque 3- moves upwards in a direction parallel to the two wires 4- rotates around an axis perpendicular to the two wires

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Problem 412

choose the letter that best answers the question or completes the statement.
1. Motion is described with respect to a b. displacement. c. slope. d. frame of reference.
2. Displacement is distance combined with a. direction. b. speed. c. velocity. d. magnitude.
3. Displacement vectors of 3 m and 5 m in the same direction combine to make a displacement vector that is a. 2 m . b. 0 m . c. 8 m . d. 15 m .
4. Average speed is the total distance divided by the a. average distance. b. average acceleration. c. total time. d. slope.
5. The slope of a distance-time graph is equal to the a. speed. b. acceleration. c. displacement. d. motion.
6. Velocity is
10. The rate at which velocity is changing at a given instant is described by (4)Text) assessment at PHSchool.com me a. instantaneous acceleration. b. average speed. c. constant speed. d. vector addition.
Understanding Concepts
11. Why is it necessary to choose a single frame reference when measuring motion?
12. For what kinds of distances would you choos make measurements in millimeters? In kilom
13. Light from a star travels to Earth in a straig line at a constant speed of almost 300,000 What is the acceleration of the light?
14. If two displacement vectors add to yield displacement of zero, what do you know the two displacements?
15. How will the total distance traveled by in 2 hours be affected if the average sp is doubled?
16. How do you know that a speedomete you the instantaneous speed of a car?
17. On a distance-time graph, what wou curve describing constant speed look
18. A spider is crawling on a wall. First it 1 meter up, then 1 meter to the left 1 meter down. What is its total disp
19. A jogger travels 8.0 kilometers in 1 What is the jogger's average spees
20. You see a lightning bolt in the sky clap of thunder 3 seconds later. travels at a speed of $330 \mathrm{~m} / \mathrm{s} . \mathrm{Hc}$ was the lightning? (Hint: Assum lightning instantly.)
7. Two or more velocities can be combined by a. graphing the slope. b. using vector addition. c. calculating the instantaneous speed. d. determining the rate.
8. A ball just dropped is an example of a. constant speed. b. instantaneous speed. c. combining displacements. d. free fall.
9. Acceleration is equal to a. distance divided by time. b. change in speed divided by time. c. the slope of a distance-time graph. d. change in speed multiplied by time.

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Problem 413

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 414

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

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Problem 415

Use the surface integral in Stokes' Theorem to calculate the circulation of the field $\mathbf{F}$ around the curve C in the indicated direction. $\mathbf{F}=y \mathbf{i}+x z \mathbf{j}+x^{2} \mathbf{k}$
C: The boundary of the triangle cut from the plane $8 x+y+z=8$ by the first octant, counterclockwise when viewed from above.
The circulation is $\square$ (Type an integer on fraction.)

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Problem 416

Question 7: Let $\left\{\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \boldsymbol{u}_{3}\right\}$ be an orthonormal basis for a three-dimensional subspace $S$ of an inner product space $V$ , and let $\boldsymbol{x}=2 \boldsymbol{u}_{1}-\boldsymbol{u}_{2}+\boldsymbol{u}_{3} \quad \text { and } \quad \boldsymbol{y}=\boldsymbol{u}_{1}+\boldsymbol{u}_{2}-4 \boldsymbol{u}_{3} .$ a) Determine the value of $\langle\boldsymbol{x}, \boldsymbol{y}\rangle$ . b) Determine the value of $\|\boldsymbol{x}\|$ .

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Problem 417

theme
Question 7 of 18 The angle $\theta$ between the vectors $u=(1,-2,3)$ and $v=(3,-1,-2)$ equal to $\cos (1 / 14)$ $\cos ^{-1}(1 / 14)$ $\cos ^{-1}(-1 / 14)$ $\cos ^{-1}(-2 / 14)$

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Problem 418

Theme
Question 9 of ${ }^{18}$ The unit vector parallel to the vector $u=(2,4,-4)$ is $(1 / 5,2 / 5,-2 / 5)$ $(1 / 9,2 / 9,-2 / 9)$ $(-1 / 3,-2 / 3,2 / 3)$ $(1 / 18,2 / 18,-2 / 18)$

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Problem 419

A vector with magnitude 4 points in a direction 295 degrees counterclockwise from the positive $x$ axis. Write the vector in component form. Vector $=$ $\square$ Give each value accurate to at least 1 decimal place Add Work
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Problem 420

Select all the correct answers.
Which vectors are unit vectors? $u=\left\langle-\frac{1}{3} \sqrt{\frac{7}{2}}, \frac{1}{3} \sqrt{\frac{13}{2}}\right\rangle$ $u=\left\langle\frac{1}{\sqrt{3}},-\sqrt{\frac{2}{3}}\right\rangle$ $u=\left\langle\frac{1}{2} \sqrt{\frac{5}{3}},-\frac{1}{2} \sqrt{\frac{7}{3}}\right\rangle$ $\mathbf{u}=\left\langle\frac{3}{\sqrt{7}},-\frac{2}{\sqrt{7}}\right\rangle$

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Problem 421

QUESTION 4
Convert the following rectangular coordinates to cylindrical coordinates. Give angles in terms of Pi. If your answer is two-thirds Pi, you would type 2 pil3. It might look familiar. Keep the square root(s) in your answer- do not use decimals. Rectangular: $(2,-2,2 s q r t 2)=$ Cylindrical: $\square$ $\square$ $\square$

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Problem 422

Exercice 2 Soit $X_{1}, X_{2}$ et $X_{3}$ , trois vecteurs de $I^{3}$ tels que : $X_{1}=(-1,5,2), X_{2}=(2,-1,2)$ et $X_{3}=(1,1,3)$ a. Calculer les combinaisons linéaires suivantes: $3 \mathrm{X}_{1}-2 \mathrm{X}_{2}+\mathrm{X}_{3} ; 3\left(\mathrm{X}_{1}-\mathrm{X}_{3}\right)+\mathrm{X}_{2}$ b. Trouver trois réels $\alpha, \beta$ et $\gamma$ non nuls, tels que $\alpha \mathrm{X}_{1}+\beta \mathrm{X}_{2}+\gamma \mathrm{X}_{3}$ ait ses deux premières composantes nulles .
Exercice 3
1. Soient $u_{1}=(1,1,1,1), u_{2}=(2,-1,2,-1), u_{3}=(4,1,4,1)$ trois vecteurs de $R^{4}$ La famille $\{\mathrm{u} 1, \mathrm{u} 2, \mathrm{u} 3\}$ est-elle libre?
2. Soient dans $\mathbb{R}^{3}$ les vecteurs $v 1=(1,1,0), v 2=(4,1,4)$ et $v 3=(2,-1,4)$ .
La famille $(v 1, v 2, v 3)$ est-elle libre ?

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Problem 423

Translating $\triangle L M N$ to the right 8 units and downward 1 unit, we get its image $\triangle L^{\prime} M^{\prime} N^{\prime}$ .
Note that $\triangle L M N$ has vertices $L(-3,6), M(-5,4)$ , and $N(-1,1)$ . Also, note that $\Delta L^{\prime} M^{\prime} N^{\prime}$ has vertices $L^{\prime}(5,5), M^{\prime}(3,3)$ , and $N^{\prime}(7,0)$ . Complete the following.

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Problem 424

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 425

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 426

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 427

Write the coordinates of the vertices after a translation 2 units left. $K^{\prime}(\square, \square)$

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Problem 428

Consider quadrilateral $P Q R S$ below.
Note that $P Q R S$ has vertices $P(-1,3), Q(4,1), R(1,-7)$ , and $S(-4,-5)$ . Complete the following to determine if $P Q R S$ is a parallelogram. (a) Find the length of $\overline{Q R}$ and the length of $\overline{P S}$ .
Give exact answers (not decimal approximations).
Length of $\overline{Q R}$ : $\square$
Length of $\overline{P S}$ : $\square$ (b) Find the length of $\overline{R S}$ and the length of $\overline{P Q}$ . Give exact answers (not decimal approximations) y
Length of $\overline{R S}$ : $\square$
Length of $\overline{P Q}$ : $\square$ (c) From parts (a) and (b), what can we conclude? The quadrilateral is a parallelogram because it has one pair of opposite sides that are congruent, even though the other two sides are not congruent. The quadrilateral is not a parallelogram because it has a pair of opposite sides that are not congruent. The quadrilateral is a parallelogram because it has two pairs of opposite sides that are congruent. It cannot be determined if the quadrilateral is a parallelogram.

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Problem 429

6 Дана прямая призма $A B C D A_{1} B_{1} C_{1} D_{1}$ . Известно, что $A B=A C=A A_{1}, \angle B A C=90^{\circ}$ . Найдите угол между прямыми $B_{1} C$ и $A B$ .

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Problem 430

这 $C(15,3)$ and $D(6,15)$ are the endpoints of a line segment. What is the midpoint $M$ of that line segment? ( ${ }^{3}$ . 7 , Write the coordinates as decimals or integers. $M=(\square, \square)$ Sulinit

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Problem 431

Which point has coordinates $\left(-4,1 \frac{1}{2}\right)$ ?
U v w

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Problem 432

Write the coordinates of the vertices after a rotation $90^{\circ}$ counterclockwise around the origin. $\begin{array}{l} J^{\prime}(\sqrt{1}, \\ K^{\prime}(, \quad, \quad) \\ L^{\prime}(\square, \square) \\ M^{\prime}(\square, \square) \end{array}$

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Problem 433

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

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Problem 434

(文) The point $U(-1,-3)$ is reflected over the $x$ -axis. What are the coordinates of the resulting point, U'? $U^{\prime}(\square, \square)$

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Problem 435

You start at ( 4,7 ). You move up 3 units and down 1 unit. Where do you end?

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Problem 436

問題 64 平面において, 原点を中心に $\frac{\pi}{6}$ 回転移動させてから, 直線 $y=-2 x$ に関して対称移動させても動かない点を求めよ。

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Problem 437

A 5.0 kg block is pushed 1.0 m at a constant velocity up a vertical wall by a constant force applied at an angle of $29.0^{\circ}$ with the horizontal, as shown in the figure.
The acceleration of gravity is $9.81 \mathrm{~m} / \mathrm{s}^{2}$ .
Drawing not to scale. If the coefficient of kinetic friction between the block and the wall is 0.30 , find a) the work done by the force on the block.
Answer in units of J .

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Problem 438

1. From the figure below, P is directed at an angle $\alpha$ from x -axis and the 200 N force is acting at a slope of 5 vertical to 12 horizontal. a. Find the value of F and $\alpha$ if $\mathrm{T}=450 \mathrm{~N}, \mathrm{P}=250 \mathrm{~N}, \beta=30^{\circ}$ , and the resultant is 300 N acting up along the y -axis. Answer: $F=484.92 \mathrm{~N}, \alpha=61.22$ 。 b. Find the value of F and $\alpha$ if $\mathrm{T}=450 \mathrm{~N}, \mathrm{P}=250 \mathrm{~N}, \beta=30^{\circ}$ and the resultant is zero. Answer: $F=264.85 \mathrm{~N}, \alpha=28.16$ 。
2. From Figure below, $P$ is directed at an angle $\alpha$ from $x$ -axis and the 200 N force is acting at a slope of 5 vertical to 12 horizontal. a. Find P and $\alpha$ if the resultant is 500 N upward to the right with a slope of 3 horizontal to 4 vertical. Answer: $P=490.68 \mathrm{~N}, \alpha=76.40$

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Problem 439

2 Consider the paralelogram formed by the points $E(1,2,3), F(0,3,5), G(0,1,2)$ and $H$ . (a) Show how to find point $H$ if this point has a z-coord inate of 4 . 1 mark (b) Find $\angle F$ using vector concepts. (c) Find the area of $\triangle E F G$ using vector concepts.

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Problem 440

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

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Problem 441

If this triangle is reflected over the line $y=k$ , what are the coordinates for $y$ ? $(-2,-6)$ $(-6.2)$ 125 (6.2)

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Problem 442

If you rotate this triangle 180 degrees counterclockwise, what are the coordinates for point $K$ $(6,-6)$ $(-6,6)$ $(-6,-6)$ $(6,6)$

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Problem 443

Determine the coordinate of the point $P(x, y)$ after a rotation of 40 degrees about $(0,0)$ , from the point $(5,0)$ . Round to 1 decimal place.

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Problem 444

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 445

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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Problem 446

Two ships leave a port. First sails at $47^{\circ}$ , 12 knots; second at $137^{\circ}$ , 14 knots. Distance apart after 1.5 hours?

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Problem 447

Find the new coordinates of the point $(2,-6)$ after applying the transformations $R_{270^{\circ}}$ and then $r_{\mathrm{y} \text{-axis}}$ .

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Problem 448

Two ships leave a port: one at $N 30^{\circ} 50^{\prime} E$ at 20.6 mph and another at $59^{\circ} 10^{\prime} E$ at 12.3 mph. Find their distance apart after 2 hours. Round to the nearest mile.

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Problem 449

إذا كان المطوح المحدد بالمتجهات

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Problem 450

Considere el plano $x-2 y+2 z=1$ y los puntos $A(-1,2,3) \quad B(1,4,4)$ que pertenecen a dicho plano. Si $A$ es el centro del cuadrad̃o y $B$ es un vértice, determine los otros vértices.

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Problem 451

Given $\vec{F}=4 \hat{i}+5 \hat{j}-6 y \hat{k}$ . Find $\oint \vec{F} \cdot \overrightarrow{d l}$ going around the loop that starts from the point $(0,0,0)$ to the point $(0,0,4)$ then to the point $(0,1,4)$ then to the point $(0,1,0)$ and back to $(0,0,0)$ . a. 4 b. -4 c. 24 d. 0 e. -24

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Problem 452

Given $\vec{F}=4 \hat{i}+5 \hat{j}-6 y \hat{k}$ . Find $\oint \vec{F} \cdot \overrightarrow{d l}$ going around the loop that starts from the point $(0,0,0)$ to the point $(0,0,4)$ then to the paint $(0,1,4)$ then to the point $(0,1,0)$ and back to $(0,0,0)$ . a. -4 b. -24 c. 0 d. 24 e. 4
Clear my choice

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1 2 3 4 5

Vector

Problem 401

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

Problem 402

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

Problem 403

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

Problem 404

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

Problem 405

Problem 406

جد قيم x التي تجعل الزاوية بين المتجهين منفرجة\text{جد قيم } \, x \, \text{التي تجعل الزاوية بين المتجهين منفرجة}جد قيم xالتي تجعل الزاوية بين المتجهين منفرجة

Problem 407

Sketch the polar point (-6, -25π/12).

Problem 408

Problem 409

\text{How do you know that the bicyclists in the first panel above are moving at a constant velocity?}

Problem 410

Relevante Lernziele: Lineare Algebra Gegeben ist ein Dreieck mit den Eckpunkten A(5,3,1),B(1,1,4)A(5,3,1), B(1,1,4)A(5,3,1),B(1,1,4) und C(4,5,4)C(4,5,4)C(4,5,4). Bestimmen Sie die Längen der Seiten und die Innenwinkel des Dreiecks.

Problem 411

Problem 412

Problem 413

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

Problem 414

Find the point that is 710\frac{7}{10}107​ of the way from A(−3,−5)A(-3,-5)A(−3,−5) to B(9,7)B(9,7)B(9,7).

Problem 415

Problem 416

Problem 417

Problem 418

Problem 419

A vector with magnitude 4 points in a direction 295 degrees counterclockwise from the positive xxx axis. Write the vector in component form. Vector === □\square□ Give each value accurate to at least 1 decimal place Add WorkSubmit Question

Problem 420

Problem 421

Problem 422

Problem 423

Problem 424

Calculate Michael Phelps' resultant velocity swimming North at 14 m/s14 \mathrm{~m/s}14 m/s against a 5 m/s5 \mathrm{~m/s}5 m/s South current.

Problem 425

Find the resultant velocity when crossing a river with a current of 15 m/s15 \mathrm{~m/s}15 m/s downstream and crossing at 2 m/s2 \mathrm{~m/s}2 m/s north.

Problem 426

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

Problem 427

Write the coordinates of the vertices after a translation 2 units left. K′(□,□)K^{\prime}(\square, \square)K′(□,□)

Problem 428

Problem 429

6 Дана прямая призма ABCDA1B1C1D1A B C D A_{1} B_{1} C_{1} D_{1}ABCDA1​B1​C1​D1​. Известно, что AB=AC=AA1,∠BAC=90∘A B=A C=A A_{1}, \angle B A C=90^{\circ}AB=AC=AA1​,∠BAC=90∘. Найдите угол между прямыми B1CB_{1} CB1​C и ABA BAB.

Problem 430

这 C(15,3)C(15,3)C(15,3) and D(6,15)D(6,15)D(6,15) are the endpoints of a line segment. What is the midpoint MMM of that line segment? ( 3{ }^{3}3. 7 , Write the coordinates as decimals or integers. M=(□,□)M=(\square, \square)M=(□,□) Sulinit

Problem 431

Which point has coordinates (−4,112)\left(-4,1 \frac{1}{2}\right)(−4,121​) ?U v w

Problem 432

Problem 433

Find the point that is 310\frac{3}{10}103​ of the way from A(−3,−6)A(-3,-6)A(−3,−6) to B(10,5)B(10,5)B(10,5).

Problem 434

(文) The point U(−1,−3)U(-1,-3)U(−1,−3) is reflected over the xxx-axis. What are the coordinates of the resulting point, U'? U′(□,□)U^{\prime}(\square, \square)U′(□,□)

Problem 435

You start at ( 4,7 ). You move up 3 units and down 1 unit. Where do you end?

Problem 436

問題 64 平面において, 原点を中心に π6\frac{\pi}{6}6π​ 回転移動させてから, 直線 y=−2xy=-2 xy=−2x に関して対称移動させても動かない点を求めよ。

Problem 437

Problem 438

Problem 439

Problem 440

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

Problem 441

If this triangle is reflected over the line y=ky=ky=k, what are the coordinates for yyy ? (−2,−6)(-2,-6)(−2,−6) (−6.2)(-6.2)(−6.2) 125 (6.2)

Problem 442

If you rotate this triangle 180 degrees counterclockwise, what are the coordinates for point KKK (6,−6)(6,-6)(6,−6) (−6,6)(-6,6)(−6,6) (−6,−6)(-6,-6)(−6,−6) (6,6)(6,6)(6,6)

Problem 443

Determine the coordinate of the point P(x,y)P(x, y)P(x,y) after a rotation of 40 degrees about (0,0)(0,0)(0,0), from the point (5,0)(5,0)(5,0). Round to 1 decimal place.

Problem 444

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

Problem 445

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?

Problem 446

Two ships leave a port. First sails at 47∘47^{\circ}47∘, 12 knots; second at 137∘137^{\circ}137∘, 14 knots. Distance apart after 1.5 hours?

Problem 447

Find the new coordinates of the point (2,−6)(2,-6)(2,−6) after applying the transformations R270∘R_{270^{\circ}}R270∘​ and then ry-axisr_{\mathrm{y} \text{-axis}}ry-axis​.

Problem 448

Two ships leave a port: one at N30∘50′EN 30^{\circ} 50^{\prime} EN30∘50′E at 20.6 mph and another at 59∘10′E59^{\circ} 10^{\prime} E59∘10′E at 12.3 mph. Find their distance apart after 2 hours. Round to the nearest mile.

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

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

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

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

$\text{جد قيم } \, x \, \text{التي تجعل الزاوية بين المتجهين منفرجة}$

Relevante Lernziele: Lineare Algebra Gegeben ist ein Dreieck mit den Eckpunkten $A(5,3,1), B(1,1,4)$ und $C(4,5,4)$ . Bestimmen Sie die Längen der Seiten und die Innenwinkel des Dreiecks.

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

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

A vector with magnitude 4 points in a direction 295 degrees counterclockwise from the positive $x$ axis. Write the vector in component form. Vector $=$ $\square$ Give each value accurate to at least 1 decimal place Add Work
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Calculate Michael Phelps' resultant velocity swimming North at $14 \mathrm{~m/s}$ against a $5 \mathrm{~m/s}$ South current.

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.

Write the coordinates of the vertices after a translation 2 units left. $K^{\prime}(\square, \square)$

6 Дана прямая призма $A B C D A_{1} B_{1} C_{1} D_{1}$ . Известно, что $A B=A C=A A_{1}, \angle B A C=90^{\circ}$ . Найдите угол между прямыми $B_{1} C$ и $A B$ .

这 $C(15,3)$ and $D(6,15)$ are the endpoints of a line segment. What is the midpoint $M$ of that line segment? ( ${ }^{3}$ . 7 , Write the coordinates as decimals or integers. $M=(\square, \square)$ Sulinit

Which point has coordinates $\left(-4,1 \frac{1}{2}\right)$ ?
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Find the point that is $\frac{3}{10}$ of the way from $A(-3,-6)$ to $B(10,5)$ .

(文) The point $U(-1,-3)$ is reflected over the $x$ -axis. What are the coordinates of the resulting point, U'? $U^{\prime}(\square, \square)$

問題 64 平面において, 原点を中心に $\frac{\pi}{6}$ 回転移動させてから, 直線 $y=-2 x$ に関して対称移動させても動かない点を求めよ。

If this triangle is reflected over the line $y=k$ , what are the coordinates for $y$ ? $(-2,-6)$ $(-6.2)$ 125 (6.2)

If you rotate this triangle 180 degrees counterclockwise, what are the coordinates for point $K$ $(6,-6)$ $(-6,6)$ $(-6,-6)$ $(6,6)$

Determine the coordinate of the point $P(x, y)$ after a rotation of 40 degrees about $(0,0)$ , from the point $(5,0)$ . Round to 1 decimal place.

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?

Two ships leave a port. First sails at $47^{\circ}$ , 12 knots; second at $137^{\circ}$ , 14 knots. Distance apart after 1.5 hours?

Find the new coordinates of the point $(2,-6)$ after applying the transformations $R_{270^{\circ}}$ and then $r_{\mathrm{y} \text{-axis}}$ .

Two ships leave a port: one at $N 30^{\circ} 50^{\prime} E$ at 20.6 mph and another at $59^{\circ} 10^{\prime} E$ at 12.3 mph. Find their distance apart after 2 hours. Round to the nearest mile.

Considere el plano $x-2 y+2 z=1$ y los puntos $A(-1,2,3) \quad B(1,4,4)$ que pertenecen a dicho plano. Si $A$ es el centro del cuadrad̃o y $B$ es un vértice, determine los otros vértices.