Chapter 1: Exercises (1.1 - 1.5)

1.1 Starting with the definition 1 in. = 2.54 cm, find the number of kilometers in 1.00 mile.

1.2 According to the label on a bottle of salad dressing, the volume of the contents is 0.437 liter (L). Using only the conversions 1 L = 1000 cm³ and 1 in. = 2.54 cm, express this volume in cubic inches.

1.3 How many nanoseconds does it take light to travel 1.00 km in a vacuum?

1.4 The density of lead is 11.3 g/cm³. What is this value in kilograms per cubic meter?

1.5 The most powerful engine available for the classic 1963 Chevrolet Corvette Sting Ray developed 360 horsepower and had a displacement of 327 cubic inches. Express this displacement in liters (L) by using only the conversions 1 L = 1000 cm³ and 1 in. = 2.54 cm.

Chapter 1: Discussion Questions (Q1.21 - Q1.23)

Q1.21 Let $\bf \overrightarrow{A}$ represent any nonzero vector. Why is $\bf \overrightarrow{A}$/A a unit vector and what is it's direction? If $\theta$ is the angle that $\bf \overrightarrow{A}$ makes with the $+x$-axis, explain why ($\bf \overrightarrow{A}$/A)$\cdot\hat{i}$ is called the direction cosine for that axis.

Response: By definition, a unit vector is vector with a magnitude of 1 and no units. Its only purpose is to point; that is, to describe a direction in space. $\bf \overrightarrow{A}$/A is a vector with a magnitude of 1 and no units (the units of the vector cancel out with the units of the magnitude). The direction of that vector is the same as $\bf \overrightarrow{A}$. ($\bf \overrightarrow{A}$/A)$\cdot\hat{i} = (1)(1)cos\theta = cos\theta$.


Q1.22 Which of the following are legitimate mathematical operations:
a) $\bf \overrightarrow{A}\cdot(\bf \overrightarrow{B}-\bf \overrightarrow{C})$;
b) $(\bf \overrightarrow{A}- \bf \overrightarrow{B})\times\bf \overrightarrow{C}$;
c) $\bf \overrightarrow{A}\cdot(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$;
d) $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$;
e) $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\cdot\bf \overrightarrow{C})$?
In each case, give the reason for your answer.

Response:
a) $(\bf \overrightarrow{B}-\bf \overrightarrow{C})$ is a vector quantity, so this is a valid operation.
b) $(\bf \overrightarrow{A}- \bf \overrightarrow{B})$ is a vector quantity, so this is a valid operation.
c) $(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$ is a vector quantity, so this is valid operation.
d) See (c).
e) $(\bf \overrightarrow{B}\cdot\bf \overrightarrow{C})$ is a scalar quantity, so this is not a valid operation.


Q1.23 Consider the two repeated vector products $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C})$ and $(\bf \overrightarrow{A}\times\bf \overrightarrow{B})\times\bf \overrightarrow{C}$. Give an example that illustrates the general rule that these two vector products do not have the same magnitude or direction. Can you choose vectors $\bf \overrightarrow{A}$, $\bf \overrightarrow{B}$, and $\bf \overrightarrow{C}$ such that these two vector products are equal to each other? If so, give an example.

Response: Take the vectors
$\bf \overrightarrow{A} = 1\hat{i} + 2\hat{j} + 0\hat{k}$
$\bf \overrightarrow{B} = 2\hat{i} + 1\hat{j} + 0\hat{k}$ and
$\bf \overrightarrow{C} = 3\hat{i} + 4\hat{j} + 0\hat{k}$

Let $\bf \overrightarrow{D} = (\bf \overrightarrow{B}\times\bf \overrightarrow{C})$. Then $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C}) = \bf \overrightarrow{A}\times\overrightarrow{D} =  \overrightarrow{E}$

Calculate the vector product $\bf \overrightarrow{D} =  0\hat{i} + 0\hat{j} + 5\hat{k}$ Then $\bf \overrightarrow{A}\times\overrightarrow{D} =  \overrightarrow{E} = 10\hat{i} - 5\hat{j} + 0\hat{k}$

Let $\bf \overrightarrow{F} = (\bf \overrightarrow{A}\times\bf \overrightarrow{B})$. Then $(\bf \overrightarrow{A}\times(\bf \overrightarrow{B})\times\bf \overrightarrow{C}) = \bf \overrightarrow{F}\times\overrightarrow{C} =  \overrightarrow{G}$

Calculate the vector product $\bf \overrightarrow{F} =  0\hat{i} + 0\hat{j} - 3\hat{k}$ Then $\bf \overrightarrow{F}\times\overrightarrow{C} =  \overrightarrow{G} = 12\hat{i} - 9\hat{j} + 0\hat{k}$

Clearly, $\bf \overrightarrow{E}\neq \overrightarrow{G}$

It is possible to choose three distinct nonzero vectors where $\bf \overrightarrow{A}\times(\bf \overrightarrow{B}\times\bf \overrightarrow{C}) = (\bf \overrightarrow{A}\times\bf \overrightarrow{B})\times\bf \overrightarrow{C}$

Choose the unit vectors $\hat{i}, \hat{j},$ and $\hat{k}$. Since $\hat{j}\times\hat{k}=\hat{i}$, and $\hat{i}\times\hat{j}=\hat{k}$,
$\hat{i}\times(\hat{j}\times\hat{k}) = \hat{i}\times\hat{i} = 0$ and $(\hat{i}\times\hat{j})\times\hat{k} = \hat{k}\times\hat{k} = 0$
$$\hat{i}\times(\hat{j}\times\hat{k}) = (\hat{i}\times\hat{j})\times\hat{k}$$

Chapter 1: Discussion Questions (Q1.16 - Q1.20)

NOTE: I learned how to implement LaTeX in this blog, so I will be using it from here on out!

Q1.16 Can you find a vector quantity that has a magnitude of zero but components that are different from zero? Explain. Can the magnitude of a vector be less than the magnitude of any of its components?

Response: The magnitude of a vector is given by $A = \sqrt{(A_x)^2+(A_y)^2+(A_z)^2}$ Since each term under the radical is squared, each term under the radical must be positive, or zero. Therefore, the radicand is zero only if all the components are zero. So a vector quantity with a magnitude of zero must have all components equal to zero as well. The magnitude must be greater than or equal to any single component. If all three components are zero, then the magnitude is zero and equal to all 3 components. If just one component is nonzero, ($A_x$, for example) then $A = \sqrt{(A_x)^2 + 0 + 0} = A_x$. A similar argument can be made for $A_y$ and $A_z$. If two components are non-zero, then $A$ must be greater than any single component, since a positive quantity is being added to the radicand.

Q1.17 (a) Does it make sense to say that a vector is negative? Why? (b) Does it make sense to say that one vector is the negative of another? Why? Does your answer here contradict what you said in part (a)?

Response: (a) It does not make sense to say that a vector is negative. That would imply a negative magnitude, but by definition a magnitude is always positive. (b) It does make sense to say that one vector is the negative of the other. The way to think of this is in terms of the additive inverse. In normal arithmetic, a number $x$ has an additive inverse, $-x$ (negative $x$) such that $x + -x = 0$. In the same vein, a vector $\bf \overrightarrow{A}$ has a vector additive inverse $\bf \overrightarrow{-A}$ such that $\bf \overrightarrow{A} + \bf \overrightarrow{-A} =$ 0 (A vector quantity with magnitude zero).


Q1.18 If $\bf \overrightarrow{C}$ is the vector sum of $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$, $\bf \overrightarrow{C} = \bf \overrightarrow{A} + \bf \overrightarrow{B}$, what must be true if C = A + B? What must be true if C = 0?

Response:  If C = A + B, then $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$ must be parallel vectors (same direction). If C = 0, then $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$ must be anti-parallel (opposite directions) and have the same magnitude.


Q1.19 If $\bf \overrightarrow{A}$ and $\bf \overrightarrow{B}$ are nonzero vectors, is it possible for $\bf \overrightarrow{A}\cdot\bf \overrightarrow{B}$ and $\bf \overrightarrow{A}\times\bf \overrightarrow{B}$ both to be zero? Explain.

Response: No. The scalar product is zero when the two vectors are perpendicular. The vector product is zero when the two vectors are parallel or anti-parallel. Two vectors cannot be both perpendicular AND parallel (or anti-parallel) in euclidean space.


Q1.20 What does $\bf \overrightarrow{A}\cdot\bf \overrightarrow{A}$, the scalar product of a vector with itself, give? What about $\bf \overrightarrow{A}\times\bf \overrightarrow{A}$, the vector product of a vector with itself?

Response: The dot product is defined as $\bf \overrightarrow{A}\cdot\bf \overrightarrow{B} = A_xB_x + A_yB_y + A_zB_z$. If $\bf \overrightarrow{B} = \bf \overrightarrow{A}$, then the dot prodct $$\bf \overrightarrow{A}\cdot\bf \overrightarrow{A} = A_xA_x + A_yA_y + A_zA_z = (A_x)^2 + (A_y)^2 +(A_z)^2.$$ The cross product $\bf \overrightarrow{C} = \bf \overrightarrow{A}\times\bf \overrightarrow{B}$ gives $C_x = A_yB_z - A_zB_y$, $C_y = A_zB_x - A_xB_z$, and $C_z = A_xB_y - A_yB_x$. If $\bf \overrightarrow{B} = \bf \overrightarrow{A}$, then the cross product $\bf \overrightarrow{C} = \bf \overrightarrow{A}\times\bf \overrightarrow{A}$ gives the following components of $\bf \overrightarrow{C}$: $$C_x = A_yA_z - A_zA_y = 0,$$ $$C_y = A_zA_x - A_xA_z = 0,$$ $$C_z = A_xA_y - A_yA_x = 0.$$ That is $\bf \overrightarrow{C} = 0$.

Chapter 1: Discussion Questions (Q1.11 - Q1.15)

Q1.11 Three archers each fire four arrows at a target. Joe's four arrows hit at points 10 cm above, 10 cm below, 10 cm to the left and 10 cm to the right of the center of the target. All four of Moe's arrows hit within 1 cm of a point 20 cm from the center, and Flo's four arrows all hit within 1 cm of the center. The contest judge says that one of the archers is precise but not accurate, another archer is accurate but not precise, and the third archer is both accurate and precise. Which description goes with which archer? Explain your reasoning.

Response: Joe is accurate but not precise (the average position of his arrows are in the bullseye, but none of the arrows are close together), Moe is precise but not accurate (all of his arrows are grouped closely together, but none are close to the bullseye), and Flo is both accurate and precise. Matt Parker has a great video explaining the difference.

Q1.12 A circular racetrack has a radius of 500 m. What is the displacement of a bicyclist when she travels around the track from the north side to the south side? When she makes one complete circle around the track? Explain your reasoning.

Response: The cyclists displacement from the northernmost point to another point on the circle is 1000 m ⋅ cos(θ), where θ is the angle formed by diameter through the north and south points on the circle (radius = 500 m, so the diameter = 1000 m) and the straight line from the northernmost point to the cyclist's current point on the circle. Note that θ must be between 0 and 90 degrees. The displacement is smallest at 90 degrees and greatest a 0 degrees. When the cyclist travels all the way around the track, the displacement is equal to zero.


Q1.13 Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero? Explain your reasoning.

Response: The sum of two vectors is zero if the two vector's magnitudes are equal. The sum of three arbitrary vectors A, B, and C if A + B = -C.


Q1.14 One sometimes speaks of the "direction of time" as evolving from past to future. Does this mean that time is a vector quantity? Explain your reasoning.

Response:  My instinct is that time is not a vector quantity. There is certainly a magnitude component, but since there is only one possible direction for time to proceed into, the direction aspect seems unnecessary/redundant. Therefore, my conclusion is that time is a scalar quantity, not a vector quantity.


Q1.15 Air traffic controllers give directions to airline pilots telling them in which direction they are to fly. These instructions are called "vectors" If these are the only instructions given, is the name "vector" used correctly? Why or why not?

Response: If these are the only instructions given, they are not true vectors because vectors must have both a magnitude and a direction. These instructions lack a magnitude (how far does the pilot need to fly)?

Chapter 1: Discussion Questions (Q1.6 - Q1.10)

Q1.6 The U.S. National Institute of Science and Technology (NIST) maintains several accurate copies of the international standard kilogram. Even after careful cleaning, these national standard kilograms are gaining mass at an average rate of about 1μg/y (1 y = 1 year) when compared every 10 years or so to the standard international kilogram. Does this apparent change have any importance? Explain.

Response: The kilogram is the only fundamental constant not based on a universal constant. The second (time) is defined as the time required for 9,129,631,770 cycles microwave radiation of a cesium atom, and the meter (distance) is based on the distance that light travels in a vacuum in 1/299,792,458 of a second.  If the mass of the kilogram standard changes, then the mass of actual kilogram changes as well. There is a Veritasium video about fixing the mass of a kilogram to a new standard involving a sphere of a specific isotope of silicon.


Q1.7 What physical phenomena (other than a pendulum or a cesium clock) could you use to define a time standard?

Response: We need a very regular periodic natural phenomena. A day is very regular (although it does change over a long timescale). Sundials are among the first types of clocks developed by humans. A moon cycle as seen from Earth is called a lunation and the cycle takes exactly 29.5305882 days. People have also used water clocks and hourglasses to keep track of time (although these are never quite precise) and would not make for good standards. On a shorter timescale, the best bet is probably a pulsar. A pulsar is a highly magnetized neutron star that emits a beam of electromagnetic radiation on a very regular interval. In fact a pulsar clock has been built and uses radio signals emitted from pulsars to keep time. 


Q1.8 Describe how you could measure the thickness of a sheet of paper with an ordinary ruler.

Response: I could stack x sheets of paper until the thickness of the stack measured one centimeter. Then, the thickness of one sheet of paper would be approximately equal to (1/x) centimeters. A quick google search says that a sheet of paper is about 0.1mm thick, so 10 sheets would equal 1mm and 100 sheets would equal 1cm.


Q1.9 The quantity π = 3.14159... is a number with no dimensions since it is the ratio of two lengths. Describe two or three other geometrical or physical quantities that are dimensionless.

Response: √2 = 1.41421... This is the ratio of the hypotenuse of an isosceles right triangle to one of the legs.
(√3)/2 = 0.86602... This is the ratio of an altitude of an equilateral triangle to one of its sides.
Φ = 1.618033... This is the ratio of the diagonal of a regular pentagon to one of its sides.


Q1.10 What are the units of volume? Suppose another student tells you that a cylinder of radius r and height h has volume given by πr³h. Explain why this cannot be right.

Response: Any unit of length gives a corresponding unit of volume, namely the volume of a cube whose side has the given length. For example, a cubic centimetre (cm³) would be the volume of a cube whose sides are one centimetre (1 cm) in length. In the International System of Units (SI), the standard unit of volume is the cubic metre (m³). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = (10 cm)³ = 1000 cubic centimetres = 0.001 cubic metres, so 1 cubic metre = 1000 litres.

The expression πr³h yields a quantity with units of length raised to the 4th power, and thus cannot represent a volume (example: r = 1m, h = 2m, then πr³h = π(1m)(1m)(1m)(2m) = 2π m⁴)

Chapter 1: Discussion Questions (Q1.1 - Q1.5)

Q1.1 How many correct experiments do we need to disprove a theory? How many to prove a theory? Explain.

Response: Only one correct experiment is needed to disprove a theory, and no finite number of correct experiments can prove a theory. However, a large amount of correct experiments can be used as evidence to support a theory within a range of validity.


Q1.2 A guidebook describes the rate of climb of a mountain rail as 120 meters per kilometer. How can you express this as a number with no units?

Response: 120 meters / 1 kilometer = 120 meters / 1000 meters = 120 meters / 1000 meters = 120 / 1000 = 0.12. This is simply the slope of the mountain.


Q1.3 Suppose you are asked to compute the tangent of 5.00 meters. Is this possible? Why or why not?

Response: This is not possible. The tangent function is a trigonometric function that takes an angle as an input. Since 5.00 meters is a quantity of length, this is not a valid input.


Q1.4 A highway contractor stated that in building a bridge deck he poured 250 yards of concrete. What do you think he meant?

Response: The contractor probably meant that he had poured 250 cubic yards of concrete. This is a measurement of volume and describes the quantity of concrete that he had poured.


Q1.5 What is your height in centimeters? What is your weight in newtons?

Response: I am 5 ft 11 inches, and I weigh 216 lbs. 5 ft 11 inches = 71 inches. 1 inch = 2.540 cm, therefore 71 inches = 71(2.540 cm) = 180 cm. 1 lb = 4.448 newtons (N), so 216 lbs = 216 (4.448 N) = 961 N.