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Unformatted text preview: schopper (as55864) – Ch1HW1 – Antoniewicz – (56465)
This printout should have 36 questions.
Multiplechoice questions may continue on
the next column or page – ﬁnd all choices
before answering.
001 (part 1 of 5) 2.0 points
Consider vectors A and B such that
A = 100, −600, −700 and 1 003 (part 3 of 5) 2.0 points
Find A .
Correct answer: 927.362.
Explanation:
We follow the same procedure as for the
previous part of the problem. B = −400, 400, 600 .
Find A + B .
1. 300, −200, −300
2. 100, −300, 100
3. 300, 200, −100
4. −300, −200, −100 correct
5. 0, 100, 300
Explanation:
To add vectors, we add respective components:
Ax + Bx = 100 + (−400) = −300 ,
Ay + By = −600 + 400 = −200 , and
Az + Bz = −700 + 600 = −100 , so
A + B = Ax + B x , Ay + B y , Az + B z
= −300, −200, −100 .
002 (part 2 of 5) 2.0 points
Find A + B .
Correct answer: 374.166.
Explanation:
The magnitude of A + B is
A+B
= (Ax + Bx )2 +(Ay + By )2 +(Az + Bz )2 = (−300)2 + (−200)2 + (−100)2 = 374.166 . A=
= A2 + A2 + A2
x
y
z
(100)2 + (−600)2 + (−700)2 = 927.362 . 004 (part 4 of 5) 2.0 points
Find B .
Correct answer: 824.621.
Explanation:
We follow the same procedure again. B=
= 2
2
2
Bx + By + Bz (−400)2 + (400)2 + (600)2 = 824.621 . 005 (part 5 of 5) 2.0 points
Find A + B .
Correct answer: 1751.98.
Explanation:
Here we simply add the values we obtained
in parts 3 and 4: A + B = 927.362 + 824.621
= 1751.98 . 006 10.0 points
Consider the following ﬁgure: schopper (as55864) – Ch1HW1 – Antoniewicz – (56465) r s t
Which of the following statements about
the three vectors in the ﬁgure are correct?
List all that apply, separated by commas. and option A is true.
Following the same procedure for the remaining options, we can determine that A,
B, and E are true statements, while the others are false.
007 (part 1 of 3) 3.333 points
A unit vector v lies in the xy plane, at an angle
of 105◦ from the +x axis, with a positive y
component. What are the components of the
unit vector v = vx , ˆy , ˆz ?
ˆ
ˆvv A s=t−r 105◦ B r =t−s C r+t=s v D s+t=r
E r+s=t
Find vx .
ˆ Correct answer: A, B, E.
Explanation:
Vector subtraction can be tricky. Just as
one way to subtract scalar quantities is to add
the negative of the number being subtracted
(i.e., instead of 5 − 3 = 2 we could write
5 + (−3) = 2), we can do the same with
vectors. We start by writing down the ﬁrst
vector. For option A, the ﬁrst vector in the
subtraction is t, so we draw it: Then, we add the negative of the vector
being subtracted. When we write down −r , it
will be pointing in the opposite direction from
r. We place −r ’s starting point at the tip of t,
and from here, we are simply adding vectors:
t s Correct answer: −0.258819.
Explanation: cos 105◦
sin 105◦ 105◦ v The x component of v is given by
ˆ t vx = cos 105◦ = −0.258819.
ˆ
008 (part 2 of 3) 3.333 points
Find vy .
ˆ
Correct answer: 0.965926. −r Explanation:
The y component of v is given by
ˆ
vy = sin 105◦ = 0.965926.
ˆ Notice that the resultant vector, s, is exactly the same as the original s in the ﬁgure
we started with. So
t + (−r ) = t − r = s, 2 009 (part 3 of 3) 3.333 points
Find vz .
ˆ
Correct answer: 0. schopper (as55864) – Ch1HW1 – Antoniewicz – (56465)
Explanation:
This vector lies in the xy plane, so it has no
z component. Therefore
vz = 0.
ˆ
010 (part 1 of 6) 1.667 points
In the following ﬁgure, three vectors are represented by arrows in the xy plane. Each
square in the grid represents one meter. Determine the components of each vector (an
accuracy of 0.5 m is ﬁne), and then calculate
the magnitude of the vector. What is A ? Give your answer in units of
meters.
Correct answer: 6.18466 m.
Explanation:
We use the Pythagorean theorem:
A= = A2 + A2 + A2
x
y
z (−6 m)2 + (+1.5 m)2 + (0 m)2
= A 3 √ 38.25 m2 = 6.18466 m. B
C 012 (part 3 of 6) 1.667 points
Which choice best represents the components
of B = Bx , By , Bz ?
1. −3 m, +4.5 m, 0 m Which choice best represents the components of A = Ax , Ay , Az ?
1. −6 m, +1.5 m, 0 m correct
2. −5.5 m, +1 m, 0 m
3. +6 m, −1.5 m, 0 m
4. −6 m, −1.5 m, 0 m
5. +5.5 m, +1.5 m, 0 m
Explanation:
To ﬁnd the components of A, we just count
the units by which A stretches in the x and y
directions, letting the direction determine the
sign. A stretches 6 units along the xdirection,
and the direction is to the left. Since the units
are meters, we know that Ax = −6 m. Similarly for the y direction, A stretches 1.5 units
upward, meaning Ay = +1.5 m. Since the
vector lies in the xy plane, the third component Az = 0.
011 (part 2 of 6) 1.667 points 2. −3 m, −4 m, 0 m
3. +3 m, −4.5 m, 0 m correct
4. +3.5 m, −4.5 m, 0 m
5. +3.5 m, −4 m, 0 m
Explanation:
B stretches 3 units along the positive xdirection, so Bx = +3 m. B stretches 4.5
units downward, meaning By = −4.5 m.
Since the vector lies in the xy plane, the third
component Bz = 0.
013 (part 4 of 6) 1.667 points
What is B ? Give your answer in units of
meters.
Correct answer: 5.40833 m.
Explanation:
The procedure is the same as for part 2:
B= 2
2
2
Bx + By + Bz schopper (as55864) – Ch1HW1 – Antoniewicz – (56465)
= (+3 m)2 + (−4.5 m)2 + (0 m)2
√
= 29.25 m2
= 5.40833 m. 014 (part 5 of 6) 1.667 points
Which choice best represents the components
of C = Cx , Cy , Cz ?
1. +2.5 m, −1.5 m, 0 m
2. +2.5 m, +1.5 m, 0 m correct 4 A star is located at
S = 2 × 1010 , −3 × 1010 , 5 × 1010 .
What is R, the vector pointing from the star
to the planet?
1. R = 5 × 1010 , −1.2 × 1011 , 6 × 1010
2. R = 8 × 1010 , −9 × 1010 , 6 × 1010
3. R = 8 × 1010 , −9 × 1010 , 7 × 1010
4. R = 4 × 1010 , −7 × 1010 , 1 × 1011 correct 3. +2.5 m, +1 m, 0 m
4. +2 m, +1.5 m, 0 m
5. −2.5 m, +1.5 m, 0 m
Explanation:
C stretches 2.5 units along the positive xdirection, so Cx = +2.5 m. C stretches 1.5
units upward, meaning Cy = +1.5 m. Since
the vector lies in the xy plane, the third component Cz = 0.
015 (part 6 of 6) 1.667 points
What is C ? Give your answer in units of
meters.
Correct answer: 2.91548 m.
Explanation:
We use the Pythagorean theorem:
C=
= 2
2
2
Cx + Cy + Cz (+2.5 m)2 + (+1.5 m)2 + (0 m)2
√
= 8 . 5 m2 5. R = 4 × 1010 , −1.1 × 1011 , 9 × 1010 Explanation:
This is a vector subtraction problem. To
ﬁnd R, we subtract S − P by respective components:
Sx − Px = 2 × 1010 − (−2 × 1010 ) = 4 × 1010
Sy − Py = −3 × 1010 + 4 × 1010 = −7 × 1010
Sz − Pz = 5 × 1010 + (−5 × 1010 ) = 1 × 1011
So
R=S−P = 4 × 1010 , −7 × 1010 , 1 × 1011 . 017 (part 2 of 5) 2.0 points
What is R ?
Correct answer: 1.28452 × 1011 . Explanation:
To ﬁnd R , we use the Pythagorean theorem. = 2.91548 m.
R=
016 (part 1 of 5) 2.0 points
A planet is located at
P = −2 × 1010 , 4 × 1010 , −5 × 1010 . (4 × 1010 )2 + (−7 × 1010 )2 + (1 × 1011 )2
= 1.65 × 1022 = 1.28452 × 1011 . schopper (as55864) – Ch1HW1 – Antoniewicz – (56465)
018 (part 3 of 5) 2.0 points
For the remaining three parts of this problem,
ˆ
you will ﬁnd the components of R, the unit
vector in the direction of R. Begin by ﬁnding
ˆ
Rx .
Correct answer: 0.3114.
Explanation:
We simply divide Rx by the magnitude:
4 × 1010
ˆ x = rx =
R
= 0.3114.
1.28452 × 1011
R ˆ
by ﬁnding A and the components of A.
Find A .
Correct answer: 469.042 m/s2 .
Explanation:
To ﬁnd the magnitude of the vector A, we
use the Pythagorean theorem:
A=
= Correct answer: −0.544949. Explanation:
We divide Ry by the magnitude:
Ry
−7 × 1010
ˆ
= −0.544949.
=
Ry =
1.28452 × 1011
R 020 (part 5 of 5) 2.0 points
ˆ z.
Find R
Correct answer: 0.778499.
Explanation:
We divide Rz by the magnitude:
1 × 1011
Rz
ˆ
=
= 0.778499.
Rz =
1.28452 × 1011
R
021 (part 1 of 4) 2.5 points
Write the vector
A = 200, 300, −300 m/s2
as the product
ˆˆˆ
ˆ
A · A = A · Ax , Ay , Az A2 + A2 + A2
x
y
z (200)2 + (300)2 + (−300)2
= 019 (part 4 of 5) 2.0 points
ˆ y.
Find R 5 2.2 × 105 = 469.042 m/s2 .
022 (part 2 of 4) 2.5 points
ˆx .
Find A
Correct answer: 0.426401 m/s2 .
Explanation:
To ﬁnd the unit vector, we use the formula
Ax , Ay , Az
A
ˆ
A=
=
,
A
A
so we can use our answer from part 1 and simply divide each component by the magnitude.
ˆ
For Ax , therefore, we get
Ax
200
ˆ
Ax =
=
= 0.426401 m/s2 .
469.042 m/s2
A
023 (part 3 of 4) 2.5 points
ˆy .
Find A
Correct answer: 0.639602 m/s2 .
Explanation:
Following the procedure from part 2, we get
Ay
300
ˆ
= 0.639602 m/s2 .
=
Ay =
2
469.042 m/s
A schopper (as55864) – Ch1HW1 – Antoniewicz – (56465)
024 (part 4 of 4) 2.5 points
ˆz
Find A
Correct answer: −0.639602 m/s2 . Correct answer: −104.
Explanation:
One more time: Explanation:
Following the procedure from part 2 again,
we get
Az
−300
ˆ
Az =
= −0.639602 m/s2 .
=
469.042 m/s2
A 025 (part 1 of 3) 3.333 points
The vector
a = 0.03, −1.8, 26.0
and the scalar f = −4. Let b = f a. What are
the components of b = bx , by , bz ?
First, ﬁnd bx .
Correct answer: −0.12. Explanation:
Multiplying a vector by a scalar means multiplying each component by that scalar. So
bx = f × a x
= −4 × 0.03
= −0.12 . bz = f × a z
= −4 × 26.0
= −104 .
028 (part 1 of 4) 2.5 points
A man is standing on the roof of a building
with his head at the position
r m = 12 m, 32 m, 13 m .
He sees the top of a tree, which is at the
position
r t = −24 m, 33 m, 43 m .
What are the components of the relative position vector r tm that points from the man’s
head to the top of the tree? Start by ﬁnding
tm
rx . Answer in m.
Correct answer: −36 m. Explanation:
We simply subtract the two x components,
starting with the position where we want the
vector to point, which in this case is the tree:
m
tm
t
rx = rx − rx
= −24 m − 12 m
= −36 m . 026 (part 2 of 3) 3.333 points
Now ﬁnd by .
Correct answer: 7.2.
Explanation:
This is the same as in the ﬁrst part: Find 029 (part 2 of 4) 2.5 points
Answer in m. tm
ry . Correct answer: 1 m.
by = f × a y
= −4 × −1.8
= 7. 2 .
027 (part 3 of 3) 3.333 points
Finally, ﬁnd bz . 6 Explanation:
Same process but with the y components:
tm
t
m
ry = ry − ry
= 33 m − 32 m
= 1 m. schopper (as55864) – Ch1HW1 – Antoniewicz – (56465)
030 (part 3 of 4) 2.5 points
tm
Find rz . Answer in m. 7 21
shown in the diagram. Start by ﬁnding rx .
Answer in m. Correct answer: 30 m.
Explanation:
Same process but with the z components: r2 tm
t
m
rz = rz − rz
= 43 m − 13 m
= 30 m . 031 (part 4 of 4) 2.5 points
What is the distance from the man’s head to
the top of the tree? Answer in m.
Correct answer: 46.8722 m.
Explanation:
This distance is just the length of the vector
tm
r , which we can ﬁnd with the Pythagorean
theorem:
r tm =
= t
m
t
m
t
m
(rx − rx )2 +(ry − ry )2 +(rz − rz )2 r1 r 21 Correct answer: 10 m.
Explanation:
21
To ﬁnd rx , we just subtract the x components, starting with the coordinate from r 2 :
21
2
1
rx = rx − rx
= 13 m − 3 m
= 10 m . (−36 m)2 + (1 m)2 + (30 m)2 √
= 2197 m2
= 46.8722 m . 033 (part 2 of 3) 3.333 points
21
Find ry . Answer in m.
Correct answer: 6 m. 032 (part 1 of 3) 3.333 points
In the following ﬁgure, the position of object
1 is given by Explanation:
Same procedure as in part 1, but with the
y components: r 1 = 3 m, −3 m, 0 .
The position of object 2 is given by
r 2 = 1 3 m, 3 m, 0 .
You will calculate the components of the relative position vector giving the position of
object 2 relative to object 1. Before putting
in your answers, see whether they are consistent with the appearance of the vector
r 2 relative to 1 = r 21 = r 2 − r 1 21
2
1
ry = ry − ry
= 3 m − (−3 m)
= 6 m. 034 (part 3 of 3) 3.333 points
Which of the following choices represents the
position of object 1 relative to object 2?
1. r 12 = 10 m, 6 m, 0 schopper (as55864) – Ch1HW1 – Antoniewicz – (56465) at 45◦ angles and using trigonometry. For instance, E ’s horizontal component seems to be
about 3.5 units. If we call E ’s length L, then
we can write down the following equation: 2. r 12 = 6 m, 10 m, 0
3. r 12 = 0, −10 m, −6 m
4. r 12 = −10 m, −6 m, 0 correct
5. r 12 = −6 m, −10 m, 0 L cos 45◦ = 3.5
⇒L= Explanation:
The relative position of object 1 to object
2 is just the negative of the vector we found
in parts 1 and 2. We ﬂip the sign of each
component to ﬁnd that
r 12 = −10 m, −6 m, 0 .
035 (part 1 of 2) 5.0 points
The following ﬁgure shows several arrows representing vectors in the xy plane.
B C
E
D 8 A F G Which vectors have magnitudes equal to
the magnitude of A? List all that apply, separated by commas. (Note: for the purposes
of this problem, you may assume that if two
vectors appear to be nearly the same length,
they are exactly the same.)
Correct answer: B, C, D, E, F.
Explanation:
Magnitude refers to the length of the vector. We can easily count the number of units
along A to ﬁnd that its length is 5. Vectors B ,
D, and F are also clearly 5 units, so we know
those are correct choices. G is clearly not 5
units, so that one is incorrect. The remaining
two vectors, C and E , are also ﬁve units long.
We can determine this by noting that they are 3. 5
≈ 5.
cos 45◦ So C and E are also correct choices.
036 (part 2 of 2) 5.0 points
Which vectors are equal to A? List all that
apply, separated by commas.
Correct answer: B, F.
Explanation:
Actually being equal to A requires not only
equal magnitude, but that the vectors be
pointing in the same direction. The only
vectors that meet both of these criteria are B
and F . ...
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 Fall '08
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