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  • Calculus: Early Transcendentals
  • Vectors and the Geometry of Space

Calculus: Early Transcendentals

James Stewart

Chapter 12

Vectors and the Geometry of Space - all with Video Answers

Educators

+ 16 more educators

Section 2

Vectors

02:03

Problem 1

Are the following quantities vectors or scalars? Explain.
(a) The cost of a theater ticket
(b) The current in a river
(c) The initial flight path from Houston to Dallas
(d) The population of the world

Linda Hand
Linda Hand
Numerade Educator
01:06

Problem 2

What is the relationship between the point $ (4, 7) $ and the vector $ \langle 4, 7 \rangle $? Illustrate with a sketch.

Carson Merrill
Carson Merrill
Numerade Educator
03:10

Problem 3

Name all the equal vectors in the parallelogram shown.

Aparna Shakti
Aparna Shakti
Numerade Educator
02:20

Problem 4

Write each combination of vectors as a single vector.
(a) $ \vec{AB} + \vec{BC} $ (b) $ \vec{CD} + \vec{DB} $
(c) $ \vec{DB} - \vec{AB} $ (d) $ \vec{DC} + \vec{CA} + \vec{AB} $

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
06:14

Problem 5

Copy the vectors in the figure and use them to draw the following vectors.
(a) $ u + v $
(b) $ u + w $
(c) $ v + w $
(d) $ u - v $
(e) $ v + u + w $
(f) $ u - w - v $

Aparna Shakti
Aparna Shakti
Numerade Educator
04:28

Problem 6

Copy the vectors in the figure and use them to draw the following vectors.
(a) $ a + b $
(b) $ a - b $
(c) $ \frac{1}{2} a $
(d) $ -3b $
(e) $ a + 2b $
(f) $ 2b - a $

Carson Merrill
Carson Merrill
Numerade Educator
02:34

Problem 7

In the figure, the tip of $ c $ and the tail of $ d $ are both the midpoint of $ QR $. Express $ c $ and $ d $ in terms of $ a $ and $ b $.

Linda Hand
Linda Hand
Numerade Educator
View

Problem 8

If the vectors in the figure satisfy $ \mid u \mid = \mid v \mid = 1 $ and $ u + v + w = 0 $, what is $ \mid w \mid $?

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:10

Problem 9

Find a vector a with representation given by the directed line segment $ \vec{AB} $. Draw $ \vec{AB} $ and the equivalent representation starting at the origin.

$ A (-2, 1), B (1, 2) $

Carson Merrill
Carson Merrill
Numerade Educator
01:07

Problem 10

Find a vector a with representation given by the directed line segment $ \vec{AB} $. Draw $ \vec{AB} $ and the equivalent representation starting at the origin.

$ A (-5, -1), B (-3, 3) $

Carson Merrill
Carson Merrill
Numerade Educator
03:40

Problem 11

Find a vector a with representation given by the directed line segment $ \vec{AB} $. Draw $ \vec{AB} $ and the equivalent representation starting at the origin.

$ A (3, -1), B (2, 3) $

Chris Trentman
Chris Trentman
Numerade Educator
01:09

Problem 12

Find a vector a with representation given by the directed line segment $ \vec{AB} $. Draw $ \vec{AB} $ and the equivalent representation starting at the origin.

$ A (3, 2), B (1, 0) $

Carson Merrill
Carson Merrill
Numerade Educator
02:39

Problem 13

Find a vector a with representation given by the directed line segment $ \vec{AB} $. Draw $ \vec{AB} $ and the equivalent representation starting at the origin.

$ A (0, 3, 1), B (2, 3, -1) $

Aparna Shakti
Aparna Shakti
Numerade Educator
01:10

Problem 14

Find a vector a with representation given by the directed line segment $ \vec{AB} $. Draw $ \vec{AB} $ and the equivalent representation starting at the origin.

$ A (0, 6, -1), B (3, 4, 4) $

Carson Merrill
Carson Merrill
Numerade Educator
01:23

Problem 15

Find the sum of the given vectors and illustrate geometrically.

$ \langle -1, 4 \rangle , \langle 6, -2 \rangle $

Carson Merrill
Carson Merrill
Numerade Educator
01:23

Problem 16

Find the sum of the given vectors and illustrate geometrically.

$ \langle 3, -1 \rangle , \langle -1, 5 \rangle $

Carson Merrill
Carson Merrill
Numerade Educator
01:36

Problem 17

Find the sum of the given vectors and illustrate geometrically.

$ \langle 3, 0, 1 \rangle , \langle 0, 8, 0 \rangle $

Carson Merrill
Carson Merrill
Numerade Educator
00:05

Problem 18

Find the sum of the given vectors and illustrate geometrically.

$ \langle 1, 3, -2 \rangle , \langle 0, 0, 6 \rangle $

Carson Merrill
Carson Merrill
Numerade Educator
01:31

Problem 19

Find $ a + b , 4a + 2b , \mid a \mid $, and $ \mid a - b \mid $.

$ a = \langle -3, 4 \rangle , b = \langle 9, -1 \rangle $

Carson Merrill
Carson Merrill
Numerade Educator
01:08

Problem 20

Find $ a + b , 4a + 2b , \mid a \mid $, and $ \mid a - b \mid $.

$ a = 5i + 3j , b = -i - 2j $

Carson Merrill
Carson Merrill
Numerade Educator
01:05

Problem 21

Find $ a + b , 4a + 2b , \mid a \mid $, and $ \mid a - b \mid $.

$ a = 4i - 3j + 2k , b = 2i - 4k $

Carson Merrill
Carson Merrill
Numerade Educator
01:08

Problem 22

Find $ a + b , 4a + 2b , \mid a \mid $, and $ \mid a - b \mid $.

$ a = \langle 8, 1, -4 \rangle , b = \langle 5, -2, 1 \rangle $

Carson Merrill
Carson Merrill
Numerade Educator
02:30

Problem 23

Find a unit vector that has the same direction as the given vector.

$ \langle 6, -2 \rangle $

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
00:58

Problem 24

Find a unit vector that has the same direction as the given vector.

$ -5i + 3j - k $

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
09:30

Problem 25

Find a unit vector that has the same direction as the given vector.

$ 8i - j + 4k $

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
01:10

Problem 26

Find the vector that has the same direction as $ \langle 6, 2, -3 \rangle $ but has length 4.

Carson Merrill
Carson Merrill
Numerade Educator
01:14

Problem 27

What is the angle between the given vector and the positive direction of the x-axis?

$ i + \sqrt{3} j $

Carson Merrill
Carson Merrill
Numerade Educator
01:06

Problem 28

What is the angle between the given vector and the positive direction of the x-axis?

$ 8i + 6j $

Carson Merrill
Carson Merrill
Numerade Educator
01:12

Problem 29

If $ v $ lies in the first quadrant and makes an angle $ \frac{\pi}{3} $ with the positive x-axis and $ \mid v \mid = 4 $, find $ v $ in component form.

Carson Merrill
Carson Merrill
Numerade Educator
01:39

Problem 30

If a child pulls a sled through the snow on a level path with a force of 50 N exerted at an angle of $ 38^\circ $ above the horizontal, find the horizontal and vertical components of the force.

Carson Merrill
Carson Merrill
Numerade Educator
02:01

Problem 31

A quarterback throws a football with angle of elevation $ 40^\circ $ and speed 60 ft/s. Find the horizontal and vertical components of the velocity vector.

Madi Sousa
Madi Sousa
Numerade Educator
01:06

Problem 32

Find the magnitude of the resultant force and the angle it makes with the positive x-axis.

Carson Merrill
Carson Merrill
Numerade Educator
01:15

Problem 33

Find the magnitude of the resultant force and the angle it makes with the positive x-axis.

Carson Merrill
Carson Merrill
Numerade Educator
14:16

Problem 34

The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction $\mathrm{N} 45^{\circ} \mathrm{W}$ at a speed of $50 \mathrm{~km} / \mathrm{h}$. (This means that the direction from which the wind blows is $45^{\circ}$ west of the northerly direction.) A pilot is steering a plane in the direction $\mathrm{N} 60^{\circ} \mathrm{E}$ at an airspeed (speed in still air) of $250 \mathrm{~km} / \mathrm{h}$. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:54

Problem 35

A woman walks due west on the deck of a ship at 3 mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the woman relative to the surface of the water.

Aparna Shakti
Aparna Shakti
Numerade Educator
View

Problem 36

A crane suspends a 500-lb steel beam horizontally by support cables (with negligible weight) attached from a hook to each end of the beam. The support cables each make an angle of $ 60^\circ $ with the beam. Find the tension vector in each support cable and the magnitude of each tension.

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
09:52

Problem 37

A block-and-tackle pulley hoist is suspended in a warehouse by ropes of lengths $2 \mathrm{~m}$ and $3 \mathrm{~m}$. The hoist weighs $350 \mathrm{~N}$. The ropes, fastened at different heights, make angles of $50^{\circ}$ and $38^{\circ}$ with the horizontal. Find the tension in each rope and the magnitude of each tension.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:12

Problem 38

The tension $ T $ at each end of a chain has magnitude 25 N (see the figure). What is the weight of the chain?

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
View

Problem 39

A boatman wants to cross a canal that is 3 km wide and wants to land at a point 2 km upstream from his starting point. The current in the canal flows at 3.5 km/h and the speed of his boat is 13 km/h.
(a) In what direction should he steer?
(b) How long will the trip take?

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
02:35

Problem 40

Three forces act on an object. Two of the forces are at an angle of $ 100^\circ $ to each other and have magnitudes 25 N and 12 N. The third is perpendicular to the plane of these two forces and has magnitude 4 N. Calculate the magnitude of the force that would exactly counterbalance these three forces.

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
01:12

Problem 41

Find the unit vectors that are parallel to the tangent line to the parabola $ y = x^2 $ at the point $ (2, 4) $.

Carson Merrill
Carson Merrill
Numerade Educator
08:09

Problem 42

(a) Find the unit vectors that are parallel to the tangent line to the curve $ y = 2 \sin x $ at the point $ (\frac{\pi}{6}, 1) $.
(b) Find the unit vectors that are perpendicular to the tangent line.
(c) Sketch the curve $ y = 2 \sin x $ and the vectors in parts (a) and (b), all starting at $ (\frac{\pi}{6}, 1) $.

AL
Andrew Lebedinsky
Numerade Educator
01:08

Problem 43

If $ A $, $ B $, and $ C $ are the vertices of a triangle, find
$$ \vec{AB} + \vec{BC} + \vec{CA} $$

Carson Merrill
Carson Merrill
Numerade Educator
03:54

Problem 44

Let $ C $ be the point on the line segment $ AB $ that is twice as far from $ B $ as it is from $ A $. If $ a = \vec{OA}, b = \vec{OB}, $ and $ c = \vec{OC} $, that $ c = \frac{2}{3} a + \frac{1}{3} b $.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:29

Problem 45

(a) Draw the vectors $ a = \langle 3, 2 \rangle $, $ b = \langle 2, -1 \rangle $, and $ c = \langle 7, 1 \rangle $.
(b) Show, by means of a sketch, that there are scalars $ s $ and $ t $ such that $ c = sa + tb $.
(c) Use the sketch to estimate the values of $ s $ and $ t $.
(d) Find the exact values of $ s $ and $ t $.

Carson Merrill
Carson Merrill
Numerade Educator
02:01

Problem 46

Suppose that $ a $ and $ b $ are nonzero vectors that are not parallel and $ c $ is any vector in the plane determined by $ a $ and $ b $. Give a geometric argument to show that $ c $ can be written as $ c = sa + tb $ for suitable scalars $ s $ and $ t $. Then give an argument using components

Carson Merrill
Carson Merrill
Numerade Educator
01:07

Problem 47

If $ r = \langle x, y, z \rangle $ and $ r_0 = \langle x_0, y_0, z_0 \rangle $, describe the set of all points $ (x, y, z) $ such that $ \mid r - r_0 \mid = 1 $.

Carson Merrill
Carson Merrill
Numerade Educator
01:21

Problem 48

If $ r = \langle x, y \rangle , r_1 = \langle x_1, y_1 \rangle $, and $ r_2 = \langle x_2, y_2 \rangle $, describe the set of all points $ (x, y) $ such that $ \mid r - r_1 \mid + \mid r - r_2 \mid = k $, where $ k > \mid r_1 - r_2 \mid $.

Carson Merrill
Carson Merrill
Numerade Educator
01:11

Problem 49

Figure 16 gives a geometric demonstration of Property 2 of vectors. Use components to give an algebraic proof of this fact for the case $ n = 2 $.

Carson Merrill
Carson Merrill
Numerade Educator
01:15

Problem 50

Prove Property 5 of vectors algebraically for the case $ n = 3 $. Then use similar triangles to give a geometric proof.

Carson Merrill
Carson Merrill
Numerade Educator
02:50

Problem 51

Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

WZ
Wen Zheng
Numerade Educator
05:38

Problem 52

Suppose the three coordinate planes are all mirrored and a light ray given by the vector $ a = \langle a_1, a_2, a_3 \rangle $ first strikes the $ xz $-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by $ b = \langle a_1, -a_2, a_3 \rangle $. Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the moon, to calculate very precisely the distance from the earth to the moon.)

WZ
Wen Zheng
Numerade Educator

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