Quiz Samples for Chapter 3 General Physics I...
Transcript of Quiz Samples for Chapter 3 General Physics I...
General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
Name: Department: Student ID #:
Notice
� +2 (−1) points per correct (incorrect) answer.
� No penalty for an unanswered question.
� Fill the blank ( ) with � (8) if the statement iscorrect (incorrect).
� Textbook: Walker, Halliday, Resnick, Principlesof Physics, Tenth Edition, John Wiley & Sons(2014).
3-1 Vectors and Their Components
1. (�) A vector is a physical quantity that has bothmagnitude and direction. The most fundamentalvector is the displacement. For example, velocity,acceleration, force, linear momentum, torque, andangular momentum are all vectors.
2. (�) A scalar is a physical quantity that has onlythe magnitude. For example, mass, length, time,temperature, energy are all scalar.
3. (�) A vector is written in the following form:
~v or v.
We can either put an arrow on top of an italicletter or write the letter in bold italic.
4. (�)
Like the displacement vector, the vector−−→AB is
identified by the difference between the initialpoint A and the final point B. The starting point
A is called the tail of a vector and the destinationB is called the head of a vector. We put an arrowto the head B of the straight line that connects Aand B.
5. (�)
Two vectors a and b are equivalent,
a = b,
if one of them can be translated to be exactlyoverlapped onto the other. Here, the translationdenotes moving a vector keeping both themagnitude and the direction.
6. (�)
We denote the vector space, the set of vectors, byV . Let a and b are two vectors that are notparallel or antiparallel. Suppose that the tails ofthe two vectors are at the same point. Let a′ andb′ that are translated from a and b so that theirtails are placed at the heads of b and a,respectively. Then a, b, a′, b′ are four sides of aparallelogram. In addition,
a = a′, b = b′.
a and a′ (b and b′) are opposite sides of a singleparallelogram.
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General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
7. (�)
The sum,~a+~b or a + b,
of two vectors ~a (a) and ~b (b) is defined as follows.
� Translate b so that the tail of b meet the headof a.
� Connect the tail of a and the head of b.
� The tail of the resultant vector a + b is thatof a.
� The head of the resultant vector a + b is thatof b.
8. (�) The three non-zero vectors a, b, and a + b isalways coplanar, placed on a single plane. If aand b are not parallel or antiparallel, the threevectors always make a triangle.
9. (�) The addition of two vectors is commutative:
a + b = b + a.
10. (�) The addition of three vectors is associative:
(a + b) + c = a + (b + c).
11. (�) The magnitude |a| of a vector a is the lengthof the vector.
12. (�) A real number x (∈ R) can be multiplied to avector a. The product xa is also a vector.
� If x = 0, then xa = 0 is the zero vector(null vector).
� If x > 0, then xa is parallel to a.
� If x < 0, then xa is antiparallel (opposite) toa.
The magnitude of xa is
|xa| = |x||a|, ∀ x ∈ R, ∀ a ∈ V .
13. (�) Distributive law is effective for scalarmultiplications: For all x, y ∈ R and a, b ∈ V
(x+ y)a = xa + ya,
x(a + b) = xa + xb.
14. (�) The null vector 0 is the additive identity .
a + 0 = 0 + a = a, ∀ a ∈ V .
15. (�) The vector −a ≡ (−1)a is the additiveinverse of a.
a + (−a) = (−a) + a = 0, ∀ a ∈ V .
3-2 Unit Vectors, Adding Vectors byComponents
1. (�) Any two vectors are coplanar.
2. (�)
The scalar product a · b of two vectors a and b isdefined by
a · b = |a||b| cos∠(a, b),
where ∠(a, b) is the angle between a and b.
3. (�) The unit vector a is defined by
a =1
|a|a.
The magnitude of a is unity and a is parallel to a.
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General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
4. (�) The Cartesian coordinate axes which isalso called the rectangular coordinate systemconsists of three orthogonal coordinate axes x, y,and z.
5. (�)
The i, j, and k are the unit vectors along the x, y,and z axes, respectively. Because they areorthonormal (any two elements are orthogonaland any one is of length unity), we find that
i · i = 1, i · j = 0, i · k = 0,
j · i = 0, j · j = 1, j · k = 0,
k · i = 0, k · j = 0, k · k = 1.
6. (�) The x, y, and z components of a vector a aredefined, respectively, by
ax = a · i,ay = a · j,az = a · k.
7. (�)
If a is on the xy plane and the angle between aand i is θ, then x and y components of a vector aare defined, respectively, by
ax = a · i = |a| cos θ,
ay = a · j = |a| cos(π2 − θ
)= |a| sin θ.
The Pythagoras theorem states that
|a| =√a2x + a2y,
cos2 θ + sin2 θ = 1,
tan θ =ayax.
8. (�)
By applying the Pythagoras theorem twice, we findthat the magnitude of a vector a in threedimensions is
|a| =√a2x + a2y + a2z,
whereax = a · i,ay = a · j,az = a · k.
9. (�) If we make use of the multiplication table forthe scalar product of Cartesian unit vectors,
i · i = 1, i · j = 0, i · k = 0,
j · i = 0, j · j = 1, j · k = 0,
k · i = 0, k · j = 0, k · k = 1,
then we can find that the scalar product a · b is
a · b = axbx + ayby + azbz.
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General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
10. (�) The scalar product of a and itself is alsodenoted by
a · a = a2 = |a|2.
11. (�) In three dimensions, a point x on a planeperpendicular to the unit vector n is
(x− a) · n = 0,
where a is a point on the plane.
12. (�)
Let a be a vector and n be an arbitrary constantunit vector. The a can be decomposed into twopieces
a = a‖ + a⊥,
where a‖ is parallel to n and a⊥ is perpendicularto n. Then, we find that
a‖ = (a · n)n,
a⊥ = a− (a · n)n.
13. (�)
Angles formed by drawing lines from the ends ofthe diameter of a circle to its circumference form aright angle. This theorem can be proved in astraightforward way if we make use of the scalarproduct.
� Let a, b, and c three vectors from the centerto three points A, B, and C on a circle.
� a2 = b2 = c2 = r2, where r is the radius ofthe circle.
� Let AB be a diameter. Then b = −a.
� The two chords can be expressed as thefollowing vectors:
−→AC = c− a, (1)−−→BC = c− b = c + a. (2)
� The scalar product of the two chord vectors is−→AC ·
−−→BC = (c− a) · (c + a)
= c2 − a2
= r2 − r2 = 0.
� Thus the angle between the two chords is 90◦.
14. (�)
The unit vector n that makes the angle 45◦ withboth i and j is
n =1√2
(i + j).
The unit vector, on the xy plane, perpendicular ton is
± 1√2
(i− j).
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General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
15. (�)
Consider an arbitrary point,
x = xi + yj,
on a circle of radius r on the xy plane whose centeris at a = ax i + ay j. The vector x satisfies thefollowing constraint equation,
(x− a)2 = r2.
16. (�)
Let i′ and j′ be the vectors obtained by rotating i
and j, respectively, by an angle θ counterclockwiseon the xy plane. Then,
i′ = cos θi + sin θj,
j′ = − sin θi + cos θj.
17. (�)
Let a′ be the vector obtained by rotating a by anangle θ counterclockwise on the xy plane. Then,
a′x = ax cos θ − ay sin θ,
a′y = ax sin θ + ay cos θ.
18. (�)
The components of a constant vector a are givenby
ax = a · i,ay = a · j.
We keep the vector a invariant and rotate theframe of reference with the new Cartesian basisvectors i′ and j′ that are obtained by rotating i andj, respectively, by an angle θ counterclockwise onthe xy plane:
i′ = cos θi + sin θj,
j′ = − sin θi + cos θj.
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General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
Then, the components of the same vector in termsof the new coordinate system are given by
a′x = a · i′ = ax cos θ + ay sin θ,
a′y = a · j′ = −ax sin θ + ay cos θ.
3-3 Multiplying Vectors
1. (�) The cross product (vector product) a× bof two vectors a and b is defined by
a× b = |a||b|n sin∠(a, b),
where ∠(a, b) is the angle between a and b. n isthe unit vector normal to the plane spanned by aand b. There are two normal directions. Thedirection of n is chosen according to theright-handed-screw rule: (a) Sweep from a to bwith the fingers of your right-hand. Youroutstretched thumb indicates the direction of n.
2. (�)
The cross product is anticommutative:
b× a = −a× b.
3. (�) The cross product vanishes if a and b arecollinear.
4. (�) The i, j, and k are the unit vectors along thex, y, and z axes, respectively. Because they areorthonormal (any two elements are orthogonaland any one is of length unity), we find that
i× i = 0, i× j = k, i× k = −j,j× i = −k, j× j = 0, j× k = i,
k× i = j, k× j = −i, k× k = 0.
5. (�)
Let a and b be two sides of a triangle. Then thearea of the triangle is
S =1
2|a× b|.
6. (�)
Let a and b be two adjacent sides of aparallelogram. Then the area of the parallelogramis
S = |a× b|.
7. (�) If we make use of the multiplication table forthe cross product of Cartesian unit vectors,
i× i = 0, i× j = k, i× k = −j,j× i = −k, j× j = 0, j× k = i,
k× i = j, k× j = −i, k× k = 0,
then we can find that the cross product a× b is
a×b = (aybz−azby )i+(azbx−axbz)j+(axby−aybx)k.
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General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
8. (�)
Let a, b, and c are three adjacent sides (edges) ofa parallelepiped. Then the volume of theparallelepiped is
V = |a · (b× c)|.
9. (�) By making use of the identitysin2 θ = 1− cos2 θ, we find that
(a× b)2 = |a× b|2 = a2b2 − (a · b)2.
10. Consider a triangle ABC. The following vectorsare defined by
a =−−→BC,
b =−→CA,
c =−−→AB.
Verify the following statements.
(a) (�) a + b + c = 0.
(b) (�) a× b = b× c = c× a.
(c) (�) The following three quantities are allequal.
|a× b| = ab sinC,
|b× c| = bc sinA,
|c× a| = ca sinB,
where A = ∠(b, c), B = ∠(c,a), andC = ∠(a, b).
(d) (�) The law of sine in Euclidean geometrycan be proved immediately from the aboveidentities as
a
sinA=
b
sinB=
c
sinC.
11. Consider two unit vectors a and b that can beobtained by rotating i by angles α and β,respectively, counterclockwise.
(a) (�) a and b are expressed in terms of i and j
as
a = cos αi + sinαj, (3a)
b = cosβ i + sinβj. (3b)
(b) (�) If we take into account the angle betweena and b, we find that
a · b = cos |α− β|.
(c) (�) If we compute a · b by making use ofEq. (3), then we find that
a · b = cosα cosβ + sinα sinβ.
(d) (�) Thus we have proved the addition rule forthe cosine function by employing the scalarproduct:
cos |α− β| = cosα cosβ + sinα sinβ.
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General Physics IQuiz Samples for Chapter 3
VectorsMarch 23, 2020
12. Consider two unit vectors a and b that can beobtained by rotating i by angles α and β,respectively, counterclockwise. We assume thatα > β.
(a) (�) If we take into account the angle betweena and b, we find that
a× b = −k sin(α− β).
(b) (�) If we compute a× b by making use ofEq. (3), then we find that
a× b = k[cosα sinβ − sinα cosβ].
(c) (�) Thus we have proved the addition rule forthe sine function by employing the crossproduct:
− sin(α− β) = cosα sinβ − sinα cosβ.
13. Consider two vectors a and b whose tails are at thesame point and they make the right angle.
(a) (�) a · b = 0.
(b) (�) a, b, and c = a− b make three sides of aright triangle. Thus we can prove thePythagoras theorem by computing c2 as
c2 = a2 + b2.
14. Consider a triangle ABC and three vectors
a =−−→BC, b =
−→CA, and c =
−−→AB. Because
a + b + c = 0,
a + b = −c, (4a)
b + c = −a, (4b)
c + a = −b. (4c)
We define α = ∠CAB, β = ∠ABC, andγ = ∠BCA.
We introduce another way to prove the law ofcosine in Euclidean geometry.
(a) (�) Squaring both sides of Eq. (4), we findthat
2a · b = c2 − a2 − b2, (5a)
2b · c = a2 − b2 − c2, (5b)
2c · a = b2 − c2 − a2. (5c)
(b) (�) The scalar products in Eq. (5) can beexpressed as
a · b = ab cos(π − γ) = −ab cos γ, (6a)
b · c = bc cos(π − α) = −bc cosα, (6b)
c · a = ca cos(π − β) = −ca cosβ. (6c)
(c) (�) Thus the cosines of the angles α, β, and γare given by
cosα =b2 + c2 − a2
2bc,
cosβ =c2 + a2 − b2
2ca,
cos γ =a2 + b2 − c2
2ab.
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