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Suppose you get a text message. Your friend
tells you to go to Lobby 7 at MIT to find
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the gift they left you 7 meters from the center
of the lobby. Is that enough information to
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find the gift right away? As you can see,
there are many locations 7 meters from the
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center of the room. Don't forget that we live
in 3 dimensions, so there are actually even
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more points 7 meters away from the center
of the room. Fortunately, in this problem,
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you can ignore most of them since we don't
expect our gift to be hanging in mid air.
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Distance alone wasn't enough information.
It would have been helpful to have both the
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distance and the direction.
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This video is part of the Representations
video series. Information can be represented
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in words, through mathematical symbols, graphically,
or in 3-D models. Representations are used
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to develop a deeper and more flexible understanding
of objects, systems, and processes.
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Hi, my name is Dan Hastings and I am Dean
of Undergraduate Education and a Professor
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of Engineering Systems, and Aeronautics and
Astronautics here at MIT. Today, I'd like
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to talk to you about the utility of thinking
about displacements as vectors when trying
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to recall vector properties, and how you determine
if a physical quantity can be represented
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using vectors.
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Before watching this video, you should know
how to add and scale vectors. You should also
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understand how to decompose vectors, and how
to find perpendicular basis vectors.
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After you watch this video, you will be able
to understand the properties of vectors by
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using displacement as an example, and you
will be able to determine whether a physical
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quantity can be represented using vectors.
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Meet the vector. The vector is an object that
has both magnitude and direction. One way
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to represent a vector is with an arrow. You
have seen other algebraic representations
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of vectors as well. There are many physical
quantities that have both magnitude and direction.
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Can you think of some? Make a list of quantities
that can be described by a magnitude and direction.
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Feel free to discuss your list with other
people. We'll come back to this list at the
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end of the video.
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Pause the video here.
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In engineering, there are many physical quantities
of interest that have both magnitude and direction.
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Consider the following example: Here you see
a video of airflow over the wing of an F16
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fighter jet model in the Wright Brothers wind
tunnel at MIT. The air that flows over the
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wing has both speed and direction. The direction
is always tangent to the path of the airflow.
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We can represent the air velocity with an
arrow at each point around the wing. The length
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of the arrow represents speed, and the direction
represents the direction of motion. Such a
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collection of vectors is called a vector field.
The vector field of airflow over the wing
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creates a lift force via the Bernoulli effect.
This effect suggests that because the horizontal
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component of the airflow velocity is the same
throughout the flow field, the air flowing
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over the wing is moving faster than the air
flowing beneath the wing. This creates a difference
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in air pressure, which provides the lift force,
another physical quantity that we can represent
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with a vector. Depending on the angle of the
wing, the magnitude and direction of the lift
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force changes. Lift is just one example of
a vector quantity that is very important in
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designing aircraft. We are quite used to thinking
of forces as vectors, but do forces exhibit
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the properties necessary to be aptly represented
by vectors? Let's review the properties of
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the vector.
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To add vector b to vector a, we connect the
tail of b to the tip of a and the sum is the
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vector that connects the tail of a to the
tip of b. An important property of vector
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addition is that it is commutative. That is
a + b = b + a. You can see this visually from
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the parallelogram whose diagonal represents
both sums simultaneously. Another important
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property is that vectors can be multiplied
by real numbers, which are called scalars,
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because they have the effect of scaling the
length of the vector. Multiplying by positive
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scalars increases the length for large scalars,
and shrinks the vector for scalars less than
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one. Multiplying a vector by -1 has the effect
of making the vector point in the opposite
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direction. Another important property of vectors
is that the initial point doesn't matter.
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Any vector pointing in the same direction
with the same magnitude represents the same
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vector. To make this seem less abstract, we
can think of vector properties in terms of
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displacement.
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Suppose you walk from a point P to a point
Q. The displacement, or change is position
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from P to Q, is aptly represented by an arrow
that starts at the point P and ends at the
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point Q. Let's see how displacement motivates
the correct form of vector addition. Consider
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the following example: you start at home,
which is represented by a star on the map.
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You walk 300 meters east to get a cup of tea
before you walk southeast 500 meters to school.
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After class you walk 400 meters southwest
of your school to play tennis. Your friend,
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who lives in your apartment complex, is going
to meet you there. What vector would represent
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the displacement vector for your friend who
leaves home directly and meets you to play
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tennis? Pause the video here and discuss your
answer with someone. Answer: The vector that
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starts at your home and moves down to the
tennis court. This is interesting because
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the arrow that connects your starting location
to your ending location represents the total
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displacement from your starting point. In
other words, this vector is the sum of the
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other 3 displacement vectors. Displacement
also helps you understand vector decomposition.
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Suppose you have walked a few blocks away,
represented by the following displacement.
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To get there, you probably didn't walk through
other people's houses and yards. Your path
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more likely looked something like this. This
process of breaking a vector down into component
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parts pointing along particular directions
is completely analogous to decomposing a vector
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into components that point along perpendicular
basis vectors. When in doubt about the mathematics
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of the vector, take a moment to rephrase your
problem in terms of displacements, and see
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if your intuition can guide the mathematics.
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Now, let's go back to forces -- do they have
the vector properties that we expect them
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to? When representing physical quantities
with vectors, the quantity must have both
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magnitude and direction. But it must also
scale and add commutatively. Let's see if
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force has these properties. Force seems to
have magnitude and direction. Force also scales
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appropriately. We think of forces as being
small or large, we can increase them and decrease
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them. When we draw a free body diagram, we
are implicitly assuming that forces are vectors,
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and that they add like vectors. But how do
we know this? We do an experiment. In this
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next segment we'll see a demonstration of
how forces, do indeed, add like vectors. [Pause]
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Here you see 3 Newton Scales connected by
strings. We'll call the two strings on top
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String A and String B.
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String A is 135 degrees off of horizontal.
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String B is 45 degrees off of horizontal.
The scale reads out the magnitude of the tension
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force on each string.
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We first want to get a reading of the scales
while there is no mass added to the system.
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The scales do not have very precise measurement;
we can only guarantee the measurement to within
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.5 Newtons.
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When looking at the bottom scale, we see that
the reading is approximately -.3 Newtons.
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The tension on string A is approximately .5
Newtons, and the tension on string B is .3
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Newtons. These tension forces are due to the
weight of the bottom scale and the strings.
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So we will need to subtract these amounts
off of any reading when mass is added into
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the system to get the tension force of the
mass alone.
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Let's add a 1kg mass to the hook below the
bottom scale. The bottom scale now reads about
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9.6 Newtons. Now we look at the top two scales.
We see that the tension force on string A
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is 7.4 Newtons, and the tension on string
B is 7.5 Newtons. We want to decompose these
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forces to see if they do in fact add like
vectors. Note that Newton's second law says
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that the sum of these forces must be zero,
since our system of strings and scales is
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stationary.
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Let's start by subtracting off the readings
we got from our Newton scale system with no
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added mass to find the net tension force due
to the mass. The tension in the string A is
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7.4-.5 = 6.9 N. And the tension on string
B is 7.5-.3=7.2 N. We find that the net force
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down is 9.6-(-.3) = 9.9 Newtons. Using F=mg,
we would predict that the force due to a 1kg
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mass would be 9.8Newtons. So the fact that
we are measuring 9.9Newtons indicates that
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we have some experimental error in our measurements.
How would you use this setup to show whether
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or not forces add like vectors?
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We want to see that the forces sum to zero.
To do this, let's decompose the forces into
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horizontal and vertical components. We use
the fact that the string A is at a 135 degree
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angle. Because the magnitude of sin(135 )and
cos(135) are both one over square root of
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2, we simply need to divide by the square
root of two. We find that the horizontal and
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vertical components of this tension force
are approximately 4.9 Newtons. Because sin(45)
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and cos(45) are both one over the square root
of two, we divide by the square root of two
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and find that each component is approximately
5.1 N. Thus the horizontal forces subtract
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to give a net force of .2N in the positive
x direction.
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The three vertical components add to 10-9.9
= .1 in the positive y direction. Because
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.1 and .2 are small, and because we know that
there are errors associated with the limits
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of accuracy of out measurements, we can be
confident that these forces do, in fact, sum
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to 0.
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So this demo does in fact suggest that forces
add like vectors. But we want to make sure
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that this wasn't an artifact of having so
much symmetry in the system. To do this, we
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move string B 60 degrees off of horizontal.
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As you can see, the tension force on the bottom
string did not change. It still reads 9.6N.
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But the force on each upper Newton scale has
changed. The tension force on String A is
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5.5N, and the tension force on the string
B is 7.5N
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We leave it as an exercise to you to decompose
the tension forces into horizontal and vertical
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components and verify that within the expected
measurement error the forces sum to zero.
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Forces really do add like vectors!!
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Okay, so forces can indeed be represented
with vectors. Let's look back at the list
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you generated of physical quantities with
both magnitude and direction. If force was
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on your list, we now know that force is indeed
a vector quantity. Maybe you also listed rotation.
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Let's see if rotations have vector properties.
Rotation seems like a physical quantity that
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has magnitude and direction. The direction
could be determined by the axis of rotation.
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We choose which way the arrow points based
on the right hand rule. The magnitude determines
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how many radians through which the object
rotates. Consider the following two rotations:
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Rz rotates an object by 90 degrees about the
z-axis: which rotates an object as such. Ry
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rotates an object by 90 degrees or π/2 radians
about the y-axis, which rotates the same object
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in this manner. Do rotations scale like vectors?
Let's see what happens if we take the vector
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that represents a rotation of π/2 radians
about the z-axis and add it to itself -- it
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seems that it should be a rotation of π radians,
or 180 degrees. And this agrees with what
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we get by rotating 90 degrees about the z-axis
twice. So scaling rotations makes sense. Question:
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Do rotations add like vectors? If we rotate
the object a quarter turn about the z-axis,
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followed by a quarter turn about the y-axis,
the object ends up in the following position.
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If instead we rotate a quarter of a turn about
the y-axis, followed by a quarter turn about
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the z-axis, the object ends up in this position.
Are the ending positions the same for the
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two different permutations of rotations? [Pause]
No, they are not. This means that rotations
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don't add commutatively, but vector addition
must be commutative. So this tells us that
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we CANNOT use vectors to represent rotations.
You'll learn that it is better to use matrices
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and matrix multiplication to represent combinations
of rotations. The tricky thing is that a vector
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can be used to represent the rotation rate,
the time derivative of rotation, quite well.
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During this video, you came up with several
physical quantities that you theorized behave
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like vectors. Consider the following list
of quantities. Compare our list to your own
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list, and determine whether each one is best
represented by a vector, a scalar, or neither.
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You may need to design an experiment or thought
experiment in order to verify your hypothesis.
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To review, you have learned that:
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Displacements help guide our intuition for
vector algebra.
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Physical quantities can be represented with
vectors only when they have magnitude, have
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direction, scale, and add commutatively.
Forces can be represented by vectors, while
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rotations cannot.