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Conceptual questions

Which of the following is a vector: a person’s height, the altitude on Mt. Everest, the age of the Earth, the boiling point of water, the cost of this book, the Earth’s population, the acceleration of gravity?

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Give a specific example of a vector, stating its magnitude, units, and direction.

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What do vectors and scalars have in common? How do they differ?

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Two campers in a national park hike from their cabin to the same spot on a lake, each taking a different path, as illustrated below. The total distance traveled along Path 1 is 7.5 km, and that along Path 2 is 8.2 km. What is the final displacement of each camper?

At the southwest corner of the figure is a cabin and in the northeast corner is a lake. A vector S with a length five point zero kilometers connects the cabin to the lake at an angle of 40 degrees north of east. Two winding paths labeled Path 1 and Path 2 represent the routes travelled from the cabin to the lake.
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If an airplane pilot is told to fly 123 km in a straight line to get from San Francisco to Sacramento, explain why he could end up anywhere on the circle shown in [link] . What other information would he need to get to Sacramento?

A map of northern California with a circle with a radius of one hundred twenty three kilometers centered on San Francisco. Sacramento lies on the circumference of this circle in a direction forty-five degrees north of east from San Francisco.
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Suppose you take two steps A and B (that is, two nonzero displacements). Under what circumstances can you end up at your starting point? More generally, under what circumstances can two nonzero vectors add to give zero? Is the maximum distance you can end up from the starting point A + B the sum of the lengths of the two steps?

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Explain why it is not possible to add a scalar to a vector.

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If you take two steps of different sizes, can you end up at your starting point? More generally, can two vectors with different magnitudes ever add to zero? Can three or more?

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Problems&Exercises

Use graphical methods to solve these problems. You may assume data taken from graphs is accurate to three digits.

Find the following for path A in [link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

A map of city is shown. The houses are in form of square blocks of side one hundred and twenty meters each. The path of A extends to three blocks towards north and then one block towards east. It is asked to find out the total distance traveled the magnitude and the direction of the displacement from start to finish.
The various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

(a) 480 m size 12{"480 m"} {}

(b) 379 m size 12{"379 m"} {} , 18.4º size 12{"18" "." "4º east of north"} {} east of north

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Find the following for path B in [link] : (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

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Find the north and east components of the displacement for the hikers shown in [link] .

north component 3.21 km, east component 3.83 km

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Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A size 12{A} {} and B size 12{B} {} , as in [link] , then this problem asks you to find their sum R = A + B size 12{"R = A + B"} {} .)

In this figure coordinate axes are shown. Vector A from the origin towards the negative of x axis is shown. From the head of the vector A another vector B is drawn towards the positive direction of y axis. The resultant R of these two vectors is shown as a vector from the tail of vector A to the head of vector B. This vector R is inclined at an angle theta with the negative x axis.
The two displacements A size 12{A} {} and B size 12{B} {} add to give a total displacement R size 12{R} {} having magnitude R size 12{R} {} and direction θ size 12{θ} {} .
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Suppose you first walk 12.0 m in a direction 20º size 12{"20" { size 12{°} } } {} west of north and then 20.0 m in a direction 40.0º size 12{"40" { size 12{°} } } {} south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A size 12{A} {} and B size 12{B} {} , as in [link] , then this problem finds their sum R = A + B size 12{ bold "R = A + B"} {} .)

In the given figure coordinates axes are shown. Vector A with tail at origin is inclined at an angle of twenty degrees with the positive direction of x axis. The magnitude of vector A is twelve meters. Another vector B is starts from the head of vector A and inclined at an angle of forty degrees with the horizontal. The resultant R of the vectors A and B is also drawn from the tail of vector A to the head of vector B. The inclination of vector R is theta with the horizontal.

19 . 5 m size 12{"19" "." "5 m"} {} , 4 . 65º size 12{4 "." "65°"} {} south of west

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Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg B size 12{B} {} , which is 20.0 m in a direction exactly 40º size 12{"20" { size 12{°} } } {} south of west, and then leg A size 12{A} {} , which is 12.0 m in a direction exactly 20º size 12{"20" { size 12{°} } } {} west of north. (This problem shows that A + B = B + A size 12{A+B=B+A} {} .)

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(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction 40.0º size 12{"40.0" { size 12{°} } } north of east (which is equivalent to subtracting B size 12{B} {} from A size 12{A} {} —that is, to finding R = A B size 12{ bold "R'"=A - B} {} ). (b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction 40.0º size 12{"40.0" { size 12{°} } } south of west and then 12.0 m in a direction 20.0º size 12{"20.0" { size 12{ ° } } } {} east of south (which is equivalent to subtracting A size 12{A} {} from B size 12{B} {} —that is, to finding R ′′ = B - A = - R size 12{R''= B – A = -R' } {} ). Show that this is the case.

(a) 26 . 6 m size 12{"26" "." "6 m"} {} , 65 . size 12{"65" "." "1º"} {} north of east

(b) 26 . 6 m size 12{"26" "." "6 m"} {} , 65 . size 12{"65" "." "1º"} {} south of west

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Show that the order of addition of three vectors does not affect their sum. Show this property by choosing any three vectors A size 12{A} {} , B size 12{B} {} , and C size 12{C} {} , all having different lengths and directions. Find the sum A + B + C size 12{ bold "A + B + C"} {} then find their sum when added in a different order and show the result is the same. (There are five other orders in which A size 12{A} {} , B size 12{B} {} , and C size 12{C} {} can be added; choose only one.)

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Show that the sum of the vectors discussed in [link] gives the result shown in [link] .

52 . 9 m size 12{"52" "." "9 m"} {} , 90 . size 12{"90" "." "1º"} {} with respect to the x -axis.

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Find the magnitudes of velocities v A size 12{v rSub { size 8{A} } } {} and v B size 12{v rSub { size 8{B} } } {} in [link]

On the graph velocity vector V sub A begins at the origin and is inclined to x axis at an angle of twenty two point five degrees. From the head of vector V sub A another vector V sub B begins. The resultant of the two vectors, labeled V sub tot, is inclined to vector V sub A at twenty six point five degrees and to the vector V sub B at twenty three point zero degrees. V sub tot has a magnitude of 6.72 meters per second.
The two velocities v A size 12{v rSub { size 8{A} } } {} and v B size 12{v rSub { size 8{B} } } {} add to give a total v tot size 12{v rSub { size 8{"tot"} } } {} .
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Find the components of v tot size 12{v rSub { size 8{"tot"} } } {} along the x - and y -axes in [link] .

x -component 4.41 m/s

y -component 5.07 m/s

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Find the components of v tot size 12{v rSub { size 8{"tot"} } } {} along a set of perpendicular axes rotated 30º size 12{"30º"} {} counterclockwise relative to those in [link] .

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Questions & Answers

does the force in a system result in the energy transfer?
Lebatam Reply
full meaning of GPS system
Anaele Reply
how to prove that Newton's law of universal gravitation F = GmM ______ R²
Kaka Reply
sir dose it apply to the human system
Olubukola Reply
prove that the centrimental force Fc= M1V² _________ r
Kaka Reply
prove that centripetal force Fc = MV² ______ r
Kaka
how lesers can transmit information
mitul Reply
griffts bridge derivative
Ganesh Reply
below me
please explain; when a glass rod is rubbed with silk, it becomes positive and the silk becomes negative- yet both attracts dust. does dust have third types of charge that is attracted to both positive and negative
Timothy Reply
what is a conductor
Timothy
hello
Timothy
below me
why below you
Timothy
no....I said below me ...... nothing below .....ok?
dust particles contains both positive and negative charge particles
Mbutene
corona charge can verify
Stephen
when pressure increases the temperature remain what?
Ibrahim Reply
remains the temperature
betuel
what is frequency
Mbionyi Reply
define precision briefly
Sujitha Reply
CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.
MITHRA Reply
hope this helps
what's critical angle
Mahmud Reply
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
dude.....next time Google it
okay whatever
Chidalu
pls who can give the definition of relative density?
Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
Chidalu
What is momentum
aliyu Reply
mass ×velocity
Chidalu
it is the product of mass ×velocity of an object
Chidalu
how do I highlight a sentence]p? I select the sentence but get options like copy or web search but no highlight. tks. src
Sean Reply
then you can edit your work anyway you want

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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