3.0 - Introduction
Knowing “how far” or “how fast” can often be useful, but “which way” sometimes
proves even more valuable. If you have ever been lost, you understand that
direction can be the most important thing to know.
Vectors describe “how much” and “which way,” or, in the terminology of physics,
magnitude and direction. You use vectors frequently, even if you are not familiar
with the term. “Go three miles northeast” or “walk two blocks north, one block east”
are both vector descriptions. Vectors prove crucial in much of physics. For example,
if you throw a ball up into the air, you need to understand that the initial velocity of
the ball points “up” while the acceleration due to the force of gravity points “down.”
In this chapter, you will learn the fundamentals of vectors: how to write them and
how to combine them using operations such as addition and subtraction.
On the right, a simulation lets you explore vectors, in this case displacement
vectors. In the simulation, you are the pilot of a small spaceship. There are three
locations nearby that you want to visit: a refueling station, a diner, and the local
gym. To reach any of these locations, you describe the displacement vector of the
spaceship by setting its x (horizontal) and y (vertical) components. In other words, you set how far horizontally you want to travel, and how far
vertically. This is a common way to express a two-dimensional vector.
There is a grid on the drawing to help you determine these values. You, and each of the places you want to visit, are at the intersection of two
grid lines. Each square on the grid is one kilometer across in each direction. Enter the values, press GO, and the simulation will show you
traveling in a straight line í along the displacement vector í according to the values you set. See if you can reach all three places. You can do
this by entering displacement values to the nearest kilometer, like (3, 4) km. To start over at any time, press RESET.
3.1 - Scalars
Scalar: A quantity that states only an amount.
Scalar quantities state an amount: “how much” or “how many.” At the right is a picture of
a dozen eggs. The quantity, a dozen, is a scalar. Unlike vectors, there is no direction
associated with a scalar í no up or down, no left or right í just one quantity, the
amount. A scalar is described by a single number, together with the appropriate units.
Temperature provides another example of a scalar quantity; it gets warmer and colder,
but at any particular time and place there is no “direction” to temperature, only a value.
Time is another commonly used scalar.
Speed and distance are yet other scalars. A speed like 60 kilometers per hour says how
fast but not which way. Distance is a scalar since it tells you how far away something is,
but not the direction.
Scalars
Amount
Only one value
Examples of scalars
12 eggs
Temperature is –5º C
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3.2 - Vectors
Vector: A quantity specified
by both magnitude and
direction.
Vectors have both magnitude (how much) and
direction. For example, vectors can be used to supply
traveling instructions. If a pilot is told “Fly 20
kilometers due south,” she is being given a
displacement vector to follow. Its magnitude is 20
kilometers and its direction is south. Vector
magnitudes are positive or zero; it would be confusing
to tell somebody to drive negative 20 kilometers south.
A spelunker (cave explorer) uses both distance and direction to navigate.
Many of the fundamental quantities in physics are vectors. For instance, displacement,
velocity and acceleration are all vector quantities. Physicists depict vectors with arrows.
The length of the arrow is proportional to the vector’s magnitude, and the arrow points
in the direction of the vector. The horizontal vector in Concept 1 on the right represents
the displacement of a car driving from Acme to Dunsville.
You see two displacement vectors in Concept 1. The displacement vector of a drive
from Acme to Dunsville is twice as long as the displacement vector from Chester to
Dunsville. This is because the distance from Acme to Dunsville is twice that of Chester
to Dunsville.
Even if they do not begin at the same point, two vectors are equal if they have the same
magnitude and direction. For instance, the vector from Chester to Dunsville in Concept
1 represents a displacement of 100 km southeast. That vector could be moved without
changing its meaning. Perhaps it is 100 km southeast from Edwards to Frankville, as
well. A vector’s meaning is defined by its length and direction, not by its starting point.
Vectors
Magnitude and direction
Represented by arrows
Length proportional to magnitude
Now that we have introduced the concept of vectors formally, we will express vector
quantities in boldface. For instance, F represents force, v stands for velocity, and so
on. You will often see F and v, as well, representing the magnitudes of the vectors,
without boldface. Why? Because it is frequently useful to discuss the magnitude of the
force or the velocity without concerning ourselves with its direction. For instance, there
may be several equations that determine the magnitude of a vector quantity like force,
but not its direction.
It is half as far from Baker to
Chester as from Acme to
Dunsville. Describe the
displacement vector from Baker
to Chester.
Displacement: 100 km, east
3.3 - Polar notation
Polar notation: Defining a vector by its angle
and magnitude.
Polar notation is a way to specify a vector. With polar notation, the magnitude and
direction of the vector are stated separately. Three kilometers due north is an example
of polar notation. “Three kilometers” is the magnitude and “north” is the direction. The
magnitude is always stated as a positive value. Instead of using “compass” or map
directions, physicists use angles. Rather than saying “three kilometers north,” a
physicist would likely say “three kilometers directed at 90 degrees.”
The angle is most conveniently measured by placing the vector’s starting point at the
origin. The angle is then typically measured from the positive side of the x axis to the
vector. This is shown in Concept 1 to the right.
Polar notation
Magnitude and angle
Angles can be positive or negative. A positive angle indicates a counterclockwise
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direction, a negative angle a clockwise direction. For example, 90° represents a quarter
turn counterclockwise from the positive x axis. In other words, a vector with a 90°
angle points straight up. We could also specify this angle as í270°.
The radian is another unit of measurement for angles that you may have seen before.
We will use degrees to specify angles unless we specifically note that we are using
radians. (Radians do prove essential at times.)
Polar notation
v is magnitude
ș is angle
Written v = (v, ș)
Write the velocity vector of the
car in polar notation.
v = (v, ș)
v = (5 m/s, 135º)
3.4 - Vector components and rectangular notation
Rectangular notation: Defining a vector by its
components.
Often what we know, or want to know, about a particular vector is not its overall
magnitude and direction, but how far it extends horizontally and vertically. On a graph,
we represent the horizontal direction as x and the vertical direction as y. These are
called Cartesian coordinates. The x component of a vector indicates its extent in the
horizontal dimension and the y component its extent in the vertical dimension.
Rectangular notation is a way to describe a vector using the components that make up
the vector. In rectangular notation, the x and y components of a vector are written
inside parentheses. A vector that extends a units along the x axis and b units along the
y axis is written as (a, b). For instance (3, 4) is a vector that extends positive three in
the x direction and positive four in the y direction from its starting point.
Vector components and
rectangular notation
x component and y component
The components of vectors are scalars with the direction indicated by their sign: x
components point right (positive) or left (negative), and y components point up (positive)
or down (negative). You see the x and y components of a car’s velocity vector in
Concept 1 at the right, shown as “hollow” vectors. The x and y values define the vector,
as they provide direction and magnitude.
For a vector A, the x and y components are sometimes written as Ax and Ay. You see
this notation used for a velocity vector v in Equation 1 and Example 1 on the right.
Consider the car shown in Example 1 on the right. Its velocity has an x component vx of
17 m/s and a y component vy of í13 m/s. We can write the car’s velocity vector as
(17, í13) m/s.
A vector can extend in more than two dimensions: z represents the third dimension.
Sometimes z is used to represent distance toward or away from you. For instance, your
computer monitor’s width is measured in the x dimension, its height with y and your
distance from the monitor with z. If you are reading this on a computer monitor and
punch your computer screen, your fist would be moving in the z dimension. (We hope
Rectangular notation
vx is horizontal component
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we’re not the cause of any such aggressive feelings.) Three-dimensional vectors are
written as (x, y, z).
The z component can also represent altitude. A Tour de France bike racer might believe
the z dimension to be the most important as he ascends one of the competition’s
famous climbs of a mountain pass.
vy is vertical component
Written v = (vx, vy)
What is the car’s velocity vector
in rectangular notation?
v = (vx, vy) m/s
v = (17, –13) m/s
3.5 - Adding and subtracting vectors graphically
Vectors can be added and subtracted. In this section, we show how to do these
operations graphically. For instance, consider the vectors A and B shown in Concept 1
to the right. The vector labeled A + B is the sum of these two vectors.
It may be helpful to imagine that these two vectors represent displacement. A person
walks along displacement vector A and then along displacement vector B. Her initial
point is the origin, and she would end up at the point at the end of the A + B vector.
The sum represents the displacement vector from her initial to final position.
To be more specific about the addition process: We start with two vectors, A and B,
both drawn starting at the origin (0, 0). To add them, we move the vector B so it starts
at the head of A. The diagram for Equation 1 shows how the B vector has been moved
so it starts at the head of A. The sum is a vector that starts at the tail of A and ends at
the head of B.
In summary, to add two vectors, you:
1.
Place the tail of the second vector at the head of the first vector. (The order of
addition does not matter, so you can place the tail of the first vector at the head of
the second as well.)
2.
Draw a vector between the tail end of the first vector and the head of the second
vector. This vector represents the sum of two vectors.
Adding vectors A + B graphically
Move tail of B to head of A
Draw vector from tail of A to head of B
To emphasize a point: You can think of this as combining a series of vector instructions.
If someone says, “Walk positive three in the x direction and then negative two in the y
direction,” you follow one instruction and then the other. This is the equivalent of placing
one vector’s tail at the head of the other. An arrow from where you started to where you
ended represents the resulting vector. Any vector is the vector sum of its rectangular
components.
When two vectors are parallel and pointing in the same direction, adding them is
relatively simple: You just combine the two arrows to form a longer arrow. If the vectors
are parallel but pointing in opposite directions, the result is a shorter arrow (three steps
forward plus two steps back equals one step forward).
Subtracting A – B graphically
Take the opposite of B
Move it to head of A
Draw vector from tail of A to head of –B
To subtract two vectors, take the opposite of the vector that is being subtracted, and
then add. (The opposite or negative of a vector is a vector with the same magnitude but
opposite direction.) This is the same as scalar subtraction (for example 20 í 5 is the same as 20 + (í5)). To draw the opposite of a vector, draw
it with the same length but the opposite direction. In other words, it starts at the same point but is rotated 180°. The diagram for Equation 2
shows the subtraction of two vectors.
When a vector is added to its opposite, the result is the zero vector, which has zero magnitude and no direction. This is analogous to adding a
scalar number to its opposite, like adding +2 and í2 to get zero.
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3.6 - Adding and subtracting vectors by components
You can combine vectors graphically, but it may be more precise to add up their
components.
You perform this operation intuitively outside physics. If you were a dancer or a
cheerleader, you would easily understand the following choreography: “Take two steps
forward, four steps to the right and one step back.” These are vector instructions. You
can add them to determine the overall result. If asked how far forward you are after this
dance move, you would say “one step,” which is two steps forward plus one step back.
You realize that your progress forward or back is unaffected by steps to the left or right.
You correctly process left/right and forward/back separately. If a physics-oriented dance
instructor asked you to describe the results of your “dancing vector” math, you would
say, “One step forward, four steps to the right.”
You have just learned the basics of vector addition, which is reasonably straightforward:
Break the vector into its components and add each component independently. In
physics though, you concern yourself with more than dance steps. You might want to
add the vector (20, í40, 60) to (10, 50, 10). Let’s assume the units for both vectors are
meters. As with the dance example, each component is added independently. You add
the first number in each set of parentheses: 20 plus 10 equals 30, so the sum along the
x axis is 30. Then you add í40 and 50 for a total of 10 along the y axis. The sum along
the z axis is 60 plus 10, or 70. The vector sum is (30, 10, 70) meters. If following all this
in the text is hard, you can see another problem worked in Example 1 on the right.
Adding and subtracting vectors
by components
Add (or subtract) each component
separately
Although we use displacement vectors in much of this discussion since they may be the
most intuitive to understand, it is important to note that all types of vectors can be added
or subtracted. You can add two velocity vectors, two acceleration vectors, two force
vectors and so on. As illustrated in the example problem, where two velocity vectors are
added, the process is identical for any type of vector.
Vector subtraction works similarly to addition when you use components. For example,
(5, 3) minus (2, 1) equals 5 minus 2, and 3 minus 1; the result is the vector (3, 2).
Adding and subtracting vectors
by components
A + B = (Ax + Bx, Ay + By)
A – B = (Ax – Bx, Ay – By)
A, B = vectors
Ax, Ay = A components
Bx, By = B components
The boat has the velocity A in
still water. Calculate its velocity
as the sum of A and the velocity
B of the river's current.
v=A+B
v = (3, 4) m/s + (2, –1) m/s
v = (3 + 2, 4 + (–1)) m/s
v = (5, 3) m/s
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3.7 - Interactive checkpoint: vector addition
What are the number values of the
constants a and b?
Answer:
a=
,b=
3.8 - Interactive checkpoint: a jogger
A jogger breaks her workout into
three segments: jogging, sprinting
and walking. Starting at home, she
jogs a displacement vector of (a, 2a)
blocks, sprints a displacement of (3
b, b) blocks, and walks back home
with a displacement of (2, í6) blocks.
What is the vector value of her
displacement during the sprint?
Answer:
S=(
,
) blocks
3.9 - Multiplying rectangular vectors by a scalar
You can multiply vector quantities by scalar quantities. Let’s say an airplane, as shown
in Concept 1 on the right, travels at a constant velocity represented by the vector (40,
10) m/s. Let's say you know its current position and want to know where it will be if it
travels for two seconds. Time is a scalar. To calculate the displacement, multiply the
velocity vector by the time.
To multiply a vector by a scalar, multiply each component of the vector by the scalar. In
this example, (2 s)(40, 10) m/s = (80, 20) m. This is the plane’s displacement vector
after two seconds of travel.
If you wanted the opposite of this vector, you would multiply by negative one. The result
in this case would be (í40, í10) m/s, representing travel at the same speed, but in the
opposite direction.
Multiplying a rectangular vector
by a scalar
Multiply each component by scalar
Positive scalar does not affect direction
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Multiplying a rectangular vector
by a scalar
sr = (srx, sry)
s = a scalar
r = a vector
rx, ry = r components
What is the displacement d of
the plane after 5.0 seconds?
d = (5.0 s)v
d = (5.0 s) (12 m/s, 15 m/s)
d = ( (5.0 s)(12 m/s), (5.0 s)(15 m/s) )
d = (60, 75) m
3.10 - Multiplying polar vectors by a scalar
Multiplying a vector represented in polar notation by a positive scalar requires only one
multiplication operation: Multiply the magnitude of the vector by the scalar. The angle is
unchanged.
Let’s say there is a vector of magnitude 50 km with an angle of 30°. You are asked to
multiply it by positive three. This situation is shown in Example 1 to the right. Since you
are multiplying by a positive scalar, the angle stays the same at 30°, and so the answer
is 150 km at 30°.
If you multiply a vector by a negative scalar, multiply its magnitude by the absolute
value of the scalar (that is, ignore the negative sign). Then change the direction of the
vector by 180° so that it points in the opposite direction. In polar notation, since the
magnitude is always positive, you add 180° to the vector's angle to take its opposite.
The result of multiplying (50 km, 30°) by negative three is (150 km, 210°).
If adding 180° would result in an angle greater than 360°, then subtract 180° instead.
For instance, in reversing an angle of 300°, subtract 180° and express the result as
120° rather than 480°. The two results are identical, but 120° is easier to understand.
Multiplying polar vector by
positive scalar
Multiply vector's magnitude by scalar
Angle unchanged
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75
Multiplying by negative scalar
Use absolute value and reverse
direction
Multiplying by negative scalar
su = (su, ș), if s positive
su = (|s|u, ș + 180°), if s negative
s = a scalar, u = a vector
u = magnitude of vector
ș = angle of vector
What is the displacement vector
if the car travels three times as
far?
su = (su, ș)
3u = ( 3(50 km), 30º)
3u = (150 km, 30º)
3.11 - Converting vectors from polar to rectangular notation
You may find it useful at times to convert a vector expressed in polar notation to rectangular coordinates. To illustrate, what if someone gave
you these directions: “Travel 3.0 km at 35° and then 2.0 km at í15°.” You suspect you could shorten this trip by adding these two vectors and
just traveling the resultant vector, but how would you add them? To add them algebraically (as opposed to graphically), it is simpler if you first
convert both to rectangular vectors.
Converting the vectors above requires some trigonometry basics, namely sines and cosines. In short, you treat the magnitude of a vector as
the hypotenuse of a right triangle, with the x component as its horizontal leg and the y component as its vertical leg.
If you want to convert the first vector above, take 3.0 km as the hypotenuse. Then calculate the x component by multiplying 3.0 km by cos
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35°,
and the y component by multiplying 3.0 km by sin 35°. Here, x = (3.0 km)(0.82) and
y = (3.0 km)(0.57), so the vector in rectangular coordinates is (2.5, 1.7) km.
Using the same method with the other vector, 2.0 km at í15° equals (1.9, í0.52) km.
The positive x component and negative y component indicate that this vector points
down and to the right, the correct direction for a vector with an angle of í15°.
We began this section by asking you how you would add these two vectors. Our work
has made this an easier problem: (2.5, 1.7) plus (1.9, í0.52) equals (4.4, 1.2). The units
are kilometers.
The x and y components can be positive or negative. For instance, the x component
will be negative when the cosine is negative, which it is for angles between 90° and
270°. This corresponds to vectors that have an x component which points to the left.
The y component will be positive when the sine is positive (between 0° and 180°, the
vector has an upward y component) and negative when the sine is negative (between
180° and 360°, the vector has a downward y component).
Since it is easy to err, it is a good practice to compare directions and the signs of the
components. In Example 1, the negative x component is correct, since the car is
moving to the left. If we had calculated a negative y component, we have erred in our
calculations, since the car is clearly moving “up” in the positive y direction.
Converting a vector from polar to
rectangular notation
To express (r, ș) as (rx, ry)
rx = r cos ș
ry = r sin ș
r = magnitude, ș = angle
rx, rx = components of vector
What is the displacement vector
r of the car in rectangular
notation?
rx = r cos ș = (2.00 km)(cos 150º)
rx = –1.73 km
ry = r sin ș = (2.00 km)(sin 150º)
ry = 1.00 km
r = (x, y) = (–1.73, 1.00) km
3.12 - Converting vectors from rectangular to polar notation
In some counties of the United States, the main roads travel either east-west or northsouth. If you wanted to drive from one town to another, the roads might force you to
travel 40 km west and then 30 km north. On the other hand, if you had a plane, you
could fly in a straight line between the two towns, which would be a shorter distance.
You would need to know the angle at which to fly and the distance. We work this
problem out in Example 1, but before that, we review the concepts necessary to solve
the problem.
To determine the angle and distance, you need to convert from rectangular to polar
coordinates. You would use trigonometry to do so. The x and y components represent
the legs of a triangle. You need to determine the length of the hypotenuse and the angle
the hypotenuse makes with the positive x axis.
In Equation 1 on the right, you see that the Pythagorean theorem is used to calculate
the hypotenuse when the two legs are known. The magnitude of the vector (the
hypotenuse) is represented with r, and the two legs, called rx and ry here, are the
components of the vector. In the example, the distance in kilometers is the square root
of (í40 km)2 + (30 km)2. That works out to 50 km.
Converting rectangular to polar
To express (rx, ry) as (r, ș)
Now you can determine the angle, which we represent as ș. As you may recall, the
tangent function relates the base and height of a right triangle to the angle between the
hypotenuse and the base (in this case, the x axis). The angle ș is the arctangent of the
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77
ratio of the two legs of the triangle. You see this in Equation 1 also.
You need to be particularly careful with tangents and arctangents since, as Equation 2
shows, two different angles can have the same tangent. For instance, the vectors (1, 1)
and (í1, í1) point in opposite directions, but since the ratio of their components in each
case is 1, the tangent equals 1 in both cases.
To find the direction of our plane's travel, we need to know the arctangent of (30 km)/
(í40 km), the ratio of the legs of the triangle. Our calculator reported back a value of
about í37°. (Make sure you know if your calculator is set for degrees or radians!)
ș = arctan (ry/rx) rx, ry = components
of vector r = magnitude, ș = angle
Unfortunately, this answer does not make sense given the circumstances described
above. A vector at an angle of í37° would have a positive x component and a negative
y component. However, the plane's displacement vector has the opposite
characteristics: a negative x component and a positive y component.
As mentioned earlier, two angles can generate the same value for a tangent, so we
need to find the other angle. With arctangents, the “other angle” is found by adding or
subtracting 180°. Adding 180° to í37° equals 143°, and 143° describes a vector with
a negative x value and a positive y value, which is appropriate for this situation.
Converting rectangular to polar
Be sure to check for the correct
angle/quadrant
What is the plane’s displacement
r in polar notation?
r = 50 km
ș = arctan(ry /rx)
ș = arctan(30 km/–40 km)
ș = arctan(–0.75)
ș = –37° + 180° = 143°
r = (r, ș) = (50 km, 143º)
3.13 - Sample problem: driving in the desert
You are told to drive 3.50 km at 42.0°,
then drive as directed by a vector of
(4.00, í3.00) km.
What is your resulting displacement in
rectangular coordinates? In polar
notation?
You are given driving directions as two displacement vectors, one stated with polar values and the second with rectangular components. You
are asked to find the resulting displacement vector in both rectangular and polar notation.
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Draw a diagram
Variables
We use A to indicate the first vector, B for the second vector, and C for their sum.
polar notation
rectangular notation
vector A
(3.50 km, 42.0°)
(Ax, Ay)
vector B
not needed
(4.00, í3.00) km
vector sum C
(C, ș)
(Cx, Cy)
What is the strategy?
1.
Convert the first vector A to rectangular notation.
2.
Add vectors A and B by adding their components. This will give you the resulting displacement
3.
Convert C to polar notation. Check to make sure the angle is in the right quadrant.
C in rectangular notation.
Mathematics principles
Polar to rectangular conversion
rx = r cos ș
ry = r sin ș
Rectangular to polar conversion
ș = arctan(ry /rx)
Adding vectors
A + B = (Ax + Bx, Ay + By)
Step-by-step solution
We start by converting vector A to rectangular notation.
Step
Reason
1.
Ax = A cos ș
2.
Ax = (3.50 km)(cos 42.0°) enter values
3.
Ax = 2.60 km
cosine, multiplication
4.
Ay = A sin ș
y component of vector
5.
Ay = (3.50 km)(sin 42.0°) enter values
6.
Ay = 2.34 km
sine, multiplication
7.
A = (2.60, 2.34) km
combine components
x component of vector
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79
We do not need to convert B to rectangular notation, since it was given to us in that form. Now we add the vectors. This gives us the answer to
the first question.
Step
Reason
8.
C=A+B
desired vector is sum
9.
C = (Ax + Bx, Ay + By)
sum of two vectors
10. C = (2.60 + 4.00, 2.34 + (–3.00)) enter values
11. C = (6.60, –0.66) km
addition
Finally, we convert C to polar notation: magnitude and angle.
Step
Reason
12.
equation for magnitude
13.
enter values
14. C = 6.63 km
evaluate
15. ș = arctan (Cy/Cx)
equation for angle
16. ș = arctan (–0.66/6.60)
enter values
17. ș = –5.71°
evaluate
18. C = (6.60 km, –5.71°)
state magnitude and angle
Once we determined the signs of the x and y components of C in step 11, we knew C pointed down and to the right. This means a negative
angle is appropriate for the polar notation. We could also have stated this angle as 354.32° (= 360° í 5.68°).
3.14 - Sample problem: looking ahead to forces
You pull on the ball with a force of
6.10 newtons at 55.0°. Your brother
applies 9.00 newtons of force at 274°.
The ball is not moving.
What is your sister's force vector in
polar notation?
In a later chapter you will learn about the concept of a force. Force is a vector quantity that follows the laws of vectors that you have learned in
this section. The unit of force is the newton (N).
If something is neither moving nor accelerating, it means that the vector sum of the forces acting on it is zero. This principle allows us to solve
the problem, since it means the force vectors must sum to zero.
Variables
polar notation
rectangular notation
your force, A
(6.10 N, 55.0°)
(Ax, Ay)
your brother's force, B
(9.00 N, 274°)
(Bx, By)
your sister's force, C
(C, ș)
(Cx, Cy)
What is the strategy?
1.
Convert A and B to rectangular notation.
2.
Set the sum of the x components of all three vectors to zero, and solve for the x component of C.
3.
Set the sum of the y components to zero, and solve for the y component of C.
4.
Convert C to polar notation.
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Mathematics principles
Polar to rectangular conversion
rx = r cos ș
ry = r sin ș
Rectangular to polar conversion
ș = arctan(ry/rx)
Adding three vectors
A + B + C = (Ax + Bx + Cx, Ay + By + Cy)
Step-by-step solution
First, we convert vector A to rectangular notation.
Step
Reason
1.
Ax = A cos ș
2.
Ax = (6.10 N)(cos 55.0°) enter values
3.
Ax = 3.50 N
evaluate
4.
Ay = A sin ș
y component of vector
5.
Ay = (6.10 N)(sin 55.0°) enter values
6.
Ay = 5.00 N
evaluate
7.
A = (3.50, 5.00) N
combine components
x component of vector
We use steps similar to those above to convert B to rectangular notation.
Step
Reason
8.
Bx = (9.00 N)(cos 274°) enter values
9.
Bx = 0.628 N
evaluate
10. By = (9.00 N)(sin 274°) enter values
11. By = –8.98 N
evaluate
12. B = (0.628, -8.98) N
combine components
Now we add the x components of all three vectors and set the sum equal to zero. This lets us solve for the
Step
x component of vector C.
Reason
13. A + B + C = 0
vector sum is zero
14. Ax + Bx + Cx = 0
sum of the x components is zero
15. 3.50 + 0.628 + Cx = 0 enter values
16. Cx = –4.13 N
solve for Cx
Similarly, we find the y component of C.
Step
17. Ay + By + Cy = 0
Reason
sum of the y components is zero
18. 5.00 + –8.98 + Cy = 0 enter values
19. Cy = 3.98 N
solve for Cy
Copyright 2000-2010 Kinetic Books Co. Chapter 3
81
Finally, we convert this vector to polar notation: magnitude and angle.
Step
Reason
20.
equation for magnitude
21.
enter values
22. C = 5.74 N
evaluate
23. ș = arctan (Cy/Cx)
equation for angle
24. ș = arctan (3.98/–4.13)
enter values
25. ș = –43.9°
evaluate
26. ș = –43.9° + 180° = 136°
add 180°
27. C = (5.74 N, 136°)
combine magnitude and angle
Since C is your sister's force vector, she pulls with a force of 5.74 N in the direction
136°.
3.15 - Interactive checkpoint: a bum steer
Alvaro, a rancher, finds one of his
steers trapped in quicksand. He tries
to free the beast by attaching a rope
and pulling with 600 newtons of
horizontal force, to no avail. He asks
his friends for help. Benjamin ties his
rope to the steer, and pulls
horizontally with 575 N at a 20.0°
angle to Alvaro. Carlos does the
same, and pulls on the other side of
Alvaro at a í10.0° angle with 500 N of
force. Alvaro’s force is directed at 0°.
What is the magnitude of their
combined force on the steer?
Answer:
F=
N
3.16 - Unit vectors
A vector can be described by its components in rectangular notation, as with
(20, 30, 40). This describes a vector that extends 20 units in the x direction, 30 units in
the y direction, and 40 units in the z direction. This form of notation is often used, but as
your studies advance, you may use another form of notation called unit vector notation
that has much in common with rectangular notation.
You see the general equation for unit vector notation in Equation 1 on the right. In this
notation, the vector (a, b, c) is written as ai + bj + ck. The unit vectors i, j and k
have lengths equal to one and point along the x, y and z dimensions respectively. So,
ai is the product of the scalar a and the unit vector i. If a is positive, the result is a
vector of magnitude a pointing in the positive x direction. If a is negative, the vector ai
points in the negative x direction, with magnitude equal to the absolute value of a.
Unit vectors are dimensionless; there are no units associated with them. The product ai
will have the same units as a. For example, (3 m/s)i + (4 m/s)j represents a velocity
vector of three meters per second in the x direction and four meters per second in the y
direction. This can also be written as (3i + 4j) m/s.
82
Unit vectors
Represent dimensions (e.g. x, y, z)
Magnitude = one
Copyright 2000-2010 Kinetic Books Co. Chapter 3
Unit vectors
(a, b, c) = ai + bj + ck
a, b, c = vector components
i, j, k = unit vectors
The x and y components of the
boat's displacement are defined
by the two equations shown.
Express the displacement at
3.0 seconds as a vector r in unit
vector notation.
x(3.0) = 3(3.0) + 4 = 13 m
y(3.0) = 4(3.0)2 – 2(3.0) + 1= 31 m
r = (13i + 31j) m
3.17 - Interactive summary problem: back to base
In the simulation on the right, three spaceships need to reach their respective
docking stations. (The red ship docks at the red station, the yellow ship at the yellow
station and so on.) Your goal is to get them all home by calculating and entering the
displacements from each of the ships to the corresponding stations.
For each of the ships you need to use a different way of describing the
displacement vector. The red ship's displacement is specified with rectangular
coordinates. The yellow ship must be specified with polar notation. And the purple
ship's displacement is some scalar multiple of (2 km, 1 km). You need to calculate
and enter the appropriate scalar value.
There is a grid on the drawing to help you determine the displacements. Each of the
ships and docks is at the intersection of two grid lines. Each square on the grid is
one kilometer across in each direction.
To dock the ships, first calculate the displacement for each ship in rectangular
coordinates.
Since the red ship uses rectangular coordinates, enter those values to the nearest
kilometer and you are done with that vessel.
Convert the yellow ship's displacement to polar notation. Enter the magnitude to the nearest 0.1 km and the angle to the nearest degree.
Determine what scalar multiple of (2, 1) will give you the purple ship's displacement, and enter it to the nearest whole integer.
After you enter the values, press GO, and see if the three ships arrive at their docking stations. Press RESET to start over. If you have
difficulty, review the sections on converting rectangular to polar notation, and on multiplying vectors by scalars.
Copyright 2000-2010 Kinetic Books Co. Chapter 3
83
3.18 - Gotchas
Confusing the sine and cosine when converting from polar to rectangular notation. For a vector of magnitude r making an angle ș with the x
axis, the x component is r cos ș and the y component is r sin ș. It is easy to forget and incorrectly use r sin ș for the x component or r cos ș
for the y component. There is a good check, used even by distinguished professors: Use 0° or 90° to verify that you are using the correct
trigonometric function. For instance, if you thought the y component was proportional to cos ș, using 90° to check that assumption would
indicate the error, because an angle of 90° corresponds to a vertical vector, but the cosine of 90° is zero.
Selecting the wrong value of arctan when converting from rectangular to polar notation. Pay attention to the quadrant where the vector lies. For
example, (í4, í3) lies in the third quadrant, but arctan (í3/í4) = arctan (0.75), and your calculator will tell you that the value is 37º. In this case
you need to add 180º to the calculator result: The angle 217º has the same tangent and it lies in the correct quadrant.
In general, check that your answers indicate a vector that points in the direction you expect!
Stating a value as a scalar when a vector is required. This happens in physics and everyday life as well. You need to use a vector when
direction is required. Throwing a ball up is different than throwing a ball down; taking highway I-5 south is different than taking I-5 north.
Vectors always start at the origin. No, they can start at any location.
3.19 - Summary
A scalar is a quantity, such as time, temperature, or speed, which indicates only
amount.
A vector is a quantity, like velocity or displacement, which has both magnitude and
direction. Vectors are represented by arrows that indicate their direction. The
arrow’s length is proportional to the vector’s magnitude. Vectors are represented
with boldface symbols, and their magnitudes are represented with italic symbols.
One way to represent a vector is with polar notation. The direction is indicated by
the angle between the positive x axis and the vector (measured in the
counterclockwise direction). For example, a vector pointing in the negative y
direction would have a direction of 270° in polar notation. The magnitude is
expressed separately. A polar vector is expressed in the form (r, ș) where r is the
magnitude and ș is the direction angle.
Another way to represent a vector is by using rectangular notation. The vector’s x
and y components are expressed as an ordered pair of numbers (x, y). The
components of a vector A are also written as Ax and Ay.
To add vectors graphically, place the tail of one on the head of the other, then draw
a vector that goes from the free tail to the free head: The new vector is the sum. To
subtract, first take the opposite of the vector being subtracted, then add. (The
opposite of a vector has the same magnitude, but it points in the opposite direction.)
You can also add and subtract vectors by writing them in rectangular notation and
adding or subtracting the x and y components separately to find the x and y
components of the sum or difference vector.
Polar notation
v = (v, ș)
Rectangular notation
v = (vx, vy)
A + B = (Ax + Bx, Ay + By)
Converting a vector
r
rx = r cos ș
ry = r sin ș
ș = arctan (ry/ rx)
Multiplying a vector by a scalar is done differently in polar and rectangular notation. For a polar vector, multiply the magnitude by the scalar. If
the result is a negative magnitude, reverse the direction of the vector by adding (or subtracting) 180ə from the angle. For a rectangular vector,
multiply each component by the scalar. To convert from polar to rectangular notation, or from rectangular to polar, use the equations shown to
the right.
84
Copyright 2000-2010 Kinetic Books Co. Chapter 3
Chapter 3 Problems
Conceptual Problems
C.1
List three quantities that are represented by vectors.
C.2
Compare these two vectors: (5, 185°) and the negative of (5, 5°). Are they the same vector? Why or why not?
C.3
An aircraft carrier sails northeast at a speed of 6.0 knots. Its velocity vector is v. What direction and speed would a ship with
velocity vector –v have?
Yes
No
knots
i.
ii.
iii.
iv.
Northwest
Northeast
Southwest
Southeast
C.4
Can the same vector have different representations in polar notation that use different angles? Explain.
C.5
A Boston cab driver picks up a passenger at Fenway Park, drops her off at the Fleet Center. Represent this displacement
with the vector D. What is the displacement vector from the Fleet Center to Fenway Park?
Yes
D
C.6
2D
0
íD
Does the multiplication of a scalar and a vector display the commutative property? This property states that the order of
multiplication does not matter. So for example, if the multiplication of a scalar s and a vector r is commutative, then sr = rs for
all values of s and r.
Yes
C.7
No
No
A small crab fishing boat travels from Colon City on the Pacific Ocean, through the Panama Canal, to Panama City on the
Atlantic Ocean. A large cruise ship travels from Colon City to Panama City by sailing all the way around the Cape Horn at the
southern tip of South America. Do these two voyages have equal displacement vectors?
Yes
No
Section Problems
Section 0 - Introduction
0.1
Use the simulation in the interactive problem in this section to answer the following questions. Assume that the simulation is
reset before each part and give each answer in the form (x, y). (a) What is the displacement to Ed's Fuel Depot? (b) What is
the displacement to Joe's Diner? (c) What is the displacement to Silver's Gym?
(a) (
km,
(b) (
km,
km)
km)
(c) (
km,
km)
Section 1 - Scalars
1.1
The Earth has a mass of 5.97×1024 kg. The Earth's Moon has a mass of 7.35×1022 kg. How many Moons would it take to
have the same mass as the Earth?
1.2
The volume of the Earth's oceans is approximately 1.4×1018 m3. The Earth's radius is 6.4×106 m. What percentage of the
Earth, by volume, is ocean?
%
1.3
Density is calculated by dividing the mass of an object by its volume. The Sun has a mass of 1.99×1030 kg and a radius of
6.96×108 m. What is the average density of the Sun?
kg/m3
Section 3 - Polar notation
3.1
The tugboat Lawowa is returning to port for the day. It has a speed of 7.00 knots. The heading to the Lawowa's port is 31.0°
west of north. If due east is 0°, what is the tug's heading as a vector in polar notation?
(
knots,
°)
Copyright 2000-2010 Kinetic Books Co. Chapter 3 Problems
85
3.2
An analog clock has stopped. Its hands are stuck displaying the time of 10 o'clock. The hour hand is 5.0 centimeters long,
and the minute hand is 11 centimeters long. Write the position vector of the tip of the hour hand, in polar notation. Consider
3 o'clock to be 0°, and assume that the center of the clock is the origin.
3.3
What is the polar notation for a vector that points from the origin to the point (0, 3.00)?
(
(
cm,
°)
,
°)
Section 4 - Vector components and rectangular notation
4.1
(a) A hotdog vendor named Sam is walking from the southwest corner of Central park to the Empire State Building. He starts
at the intersection of 59th Street and Eighth Avenue, walks 3 blocks east to 59th Street and Fifth Avenue, and then 25 blocks
south to the Empire State Building at the Corner of 34th Street and Fifth Avenue. Write his displacement vector in rectangular
notation with units of "blocks." Orient the axes so positive y is to the north, and positive x is to the east. (b) A stockbroker
named Andrea makes the same trip in a cab that gets lost, and detours 50 blocks south to Washington Square before
reorienting and finally arriving at the Empire State Building. Is her displacement any different?
(a) (
(b) Yes
4.2
4.4
) blocks
No
A treasure map you uncovered while vacationing on the Spanish Coast reads as follows: "If me treasure ye wants, me hoard
ye'll have, just follow thee directions these. Step to the south from Brisbain's Mouth, 5 paces through the trees. Then to the
west, 10 paces ye'll quest, with mud as deep as yer knees. Then 3 paces more north, and dig straight down in the Earth, and
me treasure, take it please." What is the displacement vector from Brisbain's Mouth to the spot on the Earth above the
treasure? Consider east the positive x direction and north the positive y direction.
(
4.3
,
,
) paces
Write the vectors labeled A, B and C with
rectangular coordinates.
(a) A = (
,
)
(b) B = (
,
)
(c) C = (
,
)
Consider these four vectors:
A goes from (0, 0) to (1, 2)
B from (1, í2) to (0, 2)
C from (í2, í1) to (í3, í3)
D from (í3, 1) to (í2, 3)
(a) Draw the vectors. Then answer the next two questions. (b) Which two vectors are equal? (c) Which vector is the negative
of the two equal vectors?
(a) Submit anwer on paper.
(b)
i. A and B
ii. A and C
iii. A and D
iv. B and C
v. B and D
vi. C and D
(c)
i. A
ii. B
iii. C
iv. D
Section 5 - Adding and subtracting vectors graphically
5.1
86
Draw each of the following pairs of vectors on a coordinate system, using separate coordinate systems for parts a and b of
the question. Then, on each coordinate system, also draw the vectors –B, A
(a) A = (0, 5); B = (3, 0)
(b) A = (4, 1); B = (2, –3)
+ B, and A – B. Label all your vectors.
Copyright 2000-2010 Kinetic Books Co. Chapter 3 Problems
5.2
The Moon's orbit around the Earth is nearly circular, with an average radius of 3.8×105 kilometers from the Earth's center. It
takes about 28 days for the Moon to complete one revolution around the Earth. (a) Put Earth at the origin of a coordinate
system, and draw labeled vectors to represent the moon's position at 0, 7, 14, 21 and 28 days. (b) On a separate coordinate
system, draw four labeled vectors representing the Moon's displacement from 0 days to 7 days, 7 days to 14 days, 14 days to
21 days, and 21 days to 28 days. (c) What is the sum of the four displacement vectors you drew in part "b"?
(a) Submit answer on paper.
(b) Submit answer on paper.
(c)
5.3
Consider the following vectors:
A goes from (0, 2) to (4, 2)
B from (1, í2) to (2, 1)
C from (í1, 0) to (0, 0)
D from (í3, í5) to (2, í2)
E from (í3, í2) to (í4, í5)
F from (í1, 3) to (í3, 0)
(a) Draw the vectors. Using your sketch and your knowledge of graphical vector addition and subtraction, which vector listed
above is equal to: (b) A + B? (c) E í F? (d) í(B + C)? (e) Which two vectors sum to zero?
(a) Submit answer on paper.
(b)
i. A
ii. B
iii. C
iv. D
v. E
vi. F
(c)
i. A
ii. B
iii. C
iv. D
v. E
vi. F
(d)
i. A
ii. B
iii. C
iv. D
v. E
vi. F
(e)
i. A and D
ii. B and C
iii. B and E
iv. B and F
v. C and F
vi. E and F
5.4
The racing yacht America (USA) defeated the Aurora (England) in 1851 to win the 100 Guinea Cup. From the starting buoy
the America's skipper sailed 400 meters at an angle 45° west of north, then 250 meters at an angle 30° east of north, and
finally 350 meters at an angle 60° west of north. Draw the path of the America on a coordinate system as a set of vectors
placed tip to tail. Then draw the total displacement vector.
Section 6 - Adding and subtracting vectors by components
6.1
Add the following vectors:
(a) (12, 5) + (6, 3)
(b) (í3, 8) + (6, í2)
(c) (3, 8, í7) + (7, 2, 17)
(d) (a, b, c) + (d, e, f)
(a) (
,
(b) (
,
)
)
(c) (
,
,
)
(d)
Copyright 2000-2010 Kinetic Books Co. Chapter 3 Problems
87
6.2
Solve for the unknown variables:
(a) (a, b) + (3, 3) = (6, 7)
(b) (11, c) + (d, 2) = (−12, í3)
(c) (5, 5) + (e, 4) = (2, f)
(d) ( 4, í3) í (5, g) = (h, 2)
6.3
(a) a =
;b=
(b) c =
;d=
(c) e =
;f=
(d) g =
;h=
Solve for the unknown variables: (8, 3) + (b, 2) = (4, a).
(a) a =
(b) b =
6.4
Physicists model a magnetic field by assigning to every point in space a vector that represents the strength and direction of
the field at that point. Two magnetic fields that exist in the same region of space may be added as vectors at each point to find
the representation of their combined magnetic field. The "tesla" is the unit of magnetic field strength. At a certain point,
magnet 1 contributes a field of ( í6.4, 6.1, í3.7) tesla and magnet 2 contributes a field of (í1.1, í4.5, 8.6) tesla. What is the
combined magnetic field at this point?
(
,
,
) tesla
Section 9 - Multiplying rectangular vectors by a scalar
9.1
Perform the following calculations.
(a) 6(3, í1, 8)
(b) í3(í3, 4, í5)
(c) ía(a, b, c)
(d) í2(a, 5, c) + 6 (3, íb, 2)
(a) (
,
,
)
(b) (
,
,
)
(c)
(d)
9.2
Three vectors that are neither parallel nor antiparallel can be arranged to form a triangle if they sum to (0, 0). (a) What vector
forms a triangle with (0, 3) and (3, 0)? (b) If you multiply all three vectors by the scalar 2, do they still form a triangle? (c) What
if you multiply them by the scalar a?
,
(a) (
(b) Yes
(c) Yes
)
No
No
Section 10 - Multiplying polar vectors by a scalar
10.1
Perform the following computations. Express each vector in polar notation with a positive magnitude and an angle between 0°
and 360°.
(a) 2(4, 230°)
(b) í3(7, 20°)
(c) í4(8, 260°)
(a) (
,
(b) (
,
°)
(c) (
,
°)
°)
10.2 A chimney sweep is climbing a long ladder that leans against the side of a house. If the displacement of her feet from the
base of the ladder is given by ( 2.1 ft, 65°) when she is on the third rung, what is the displacement of her feet from the base
when she has climbed twice as far?
(
88
ft,
°)
Copyright 2000-2010 Kinetic Books Co. Chapter 3 Problems
Section 11 - Converting vectors from polar to rectangular notation
11.1 Express the following vectors in rectangular notation.
(a) A = (3.00, 25.0°)
(b) B = (17.0, 135°)
(c) C = (4.00, í185°)
(d) D = í(4.00, 185°)
(a) A = (
,
)
(b) B = (
,
)
(c) C = (
,
)
(d) D = (
,
)
11.2 Consider the polar vectors A = (5, 12°) and B = (65, 90°). Which points farther in the x direction?
A
B
11.3 In the xy plane, vector A is 6.40 cm long, and at an angle of 51.0° to the x axis. Write the vector in rectangular coordinates.
(
,
) cm
11.4 The vector A, in polar notation, is (45.0, 250°), and B is (70.0, 20.0°). What is A + B in rectangular coordinates?
(
,
)
Section 12 - Converting vectors from rectangular to polar notation
12.1 Express the following vectors in polar (r, ș) notation.
(a) T = (4.00, 4.00)
(b) U = (0, 3.50)
(c) V = (í3.00, 5.00)
(d) W = (í7.00, í9.00)
(a) T = (
,
(b) U = (
,
°)
°)
(c) V = (
,
°)
,
(d) W = (
°)
12.2 The vector F is (12.0, í3.00) in rectangular notation. What is this same vector in polar notation? Express the angle as a
positive number between 0° and 360°.
(
,
°)
12.3 An adventurous aardvark arduously ambles from (3.0, 2.0) m to (í4.0, í2.0) m. State this displacement in polar coordinates.
(
m,
°)
Section 16 - Unit vectors
16.1 A helicopter takes off from the origin with a constant velocity of (16.0i + 18.0j + 4.00k) m/s. What is its position when it
reaches an altitude of 1500 meters? Altitude is measured in the k direction.
(
i+
j+
k)m
16.2 A dune buggy is driving across the desert with constant velocity, starting at an oasis. After 1.25 hours its displacement from
the oasis is given by (60.0, 32.0) km. What is its displacement 5.00 hours later?
(
,
) km
16.3 Suppose that u = 4i í j + 6k and v = 4i + 3j + 8k. Find a vector w, so that u í 2w = v.
i+
j+
k
16.4 A vector lies in the xz plane, forms an angle of 30.0° with the x axis, and has a magnitude of 10.0. Its z component is positive.
Write this vector in unit vector notation.
i+
k
16.5 Vector v is (12.0, í3.00, í4.00). Find a vector that points in the same direction as v, but has a magnitude of 1.
(
,
,
)
16.6 Find a vector with a magnitude of 1.00 that bisects the angle between the vectors 5.00i + 11.0j and 2.00i í 1.00j. Give your
answer in rectangular coordinates.
(
,
)
Copyright 2000-2010 Kinetic Books Co. Chapter 3 Problems
89
Section 17 - Interactive summary problem: back to base
17.1 Use the information given in the interactive problem in this section to calculate the following values. (a) The displacement
vector for the red ship in rectangular notation. (b) The displacement vector for the yellow ship in polar notation. (c) The scalar
multiple for the purple ship's displacement vector. Test your answers using the simulation.
(a) (
km,
km)
(b) (
km,
°)
× (2.0 km, 1.0 km)
(c)
Additional Problems
A.1
Air traffic controllers use radar to keep track of the location of aircraft. The radar displays an aircraft's location in terms of the
compass direction from the controller to the aircraft, and the horizontal distance along the ground between the two. In
addition, the aircraft transmits its altitude to the controller automatically. Create a coordinate system by making east the
positive x axis, north the positive y axis, and altitude the z axis. An aircraft is approaching at an altitude of 5.0 km, a horizontal
distance of 35 km, and a bearing of 65° east of north. Write its position vector in rectangular coordinates.
(
A.2
,
,
) km
An air traffic controller has two aircraft on radar. The first is at an altitude of 0.500 km, a horizontal distance of 3.00 km
measured along the ground to the point directly underneath the plane, and a bearing of 115° west of north. The second
aircraft is at an altitude of 1.00 km, a horizontal distance of 8.00 km, and a bearing of 35.0° east of north. Write the
displacement vector from aircraft 1 to aircraft 2 in "cylindrical" coordinates. That is, write a 3-component vector (r, ș, z) whose
first two components are the polar coordinates of the horizontal displacement between the planes, and whose third coordinate
is the vertical displacement between the planes. Consider east to be the positive x axis and north to be the positive y axis.
(
km,
°,
km)
A.3
A bird flies 5.00 m at 50.0° and then 3.00 m at í30.0°. What is the bird's total displacement in polar notation?
A.4
Show by repeated addition that multiplying the vector A = (1, 2, 3) by 3 is the same as adding up 3 copies of the vector. What
is the final vector?
(
(
90
m,
,
°)
,
)
Copyright 2000-2010 Kinetic Books Co. Chapter 3 Problems