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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.

©2020 KPOPEEE All rights reserved. Korea University of 8



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.




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.




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).




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.




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.




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.




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.
