Description: Discuss adding vectors and illustrate with components and with geometry.
Constants | PeriodicTable
Learning Goal:
To understand how vectors may be added using geometry or by representing them with components.
Fundamentally, vectors are quantities that possess both magnitude and direction. In physics problems, it is best to think of vectors as arrows, and usually it is best to
manipulate them using components. In this problem we consider the addition of two vectors using both of these methods. We will emphasize that one method is easier to
conceptualize and the other is more suited to manipulations.
Consider adding the vectors and , which have lengths and , respectively, and where makes an angle from the direction of .
In vector notation the sum is represented by
.
Addition using geometry
Part A
Which of the following procedures will add the vectors and ?
ANSWER:
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It is equally valid to put the tail of on the arrow of ; then goes from the tail of to the arrow of .
Part B
Find , the length of , the sum of and .
Express in terms of , , and angle , using radian measure for known angles.
View Available Hint(s) (2)
ANSWER:
Put the tail of on the arrow of ; goes from the tail of to the arrow of
Put the tail of on the tail of ; goes from the arrow of to the arrow of
Put the tail of on the tail of ; goes from the arrow of to the arrow of
Calculate the magnitude as the sum of the lengths and the direction as midway between and .
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Part C
Find the angle that the vector makes with vector .
Express in terms of and any of the quantities given in the problem introduction ( , , and/or ) as well as any necessary constants. Use radian
measure for known angles. Use asin for arcsine
View Available Hint(s) (1)
ANSWER:
Addition using vector components
Part D
To manipulate these vectors using vector components, we must first choose a coordinate system. In this case choosing means specifying the angle of the x axis. The
y axis must be perpendicular to this and by convention is oriented radians counterclockwise from the x axis.
Indicate whether the following statement is true or false:
There is only one unique coordinate system in which vector components can be added.
ANSWER:
=
Also accepted:
=
Also accepted: , , ,
true
false
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Part E
The key point is that you are completely free to choose any coordinate system you want in which to manipulate the vectors. It is a matter of convenience only, and so
you must consider which orientation will simplify finding the components of the given vectors and interpreting the results in that coordinate system to get the required
answer. Considering these factors, and knowing that you are going to be required to find the length of and its angle with respect to , which of the following
orientations simplifies the calculation the most?
ANSWER:
Part F
Find the components of in the coordinate system shown.
Express your answer as an ordered pair: x component, y component ; in terms of and . Use
radian measure for known angles.
ANSWER:
The x axis should be oriented along
The x axis should be oriented along
The x axis should be oriented along
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Part G
In the same coordinate system, what are the components of ?
Express your answer as an ordered pair separated by a comma. Give your answer in terms of variables defined in the introduction ( , , and ). Use
radian measure for known angles.
ANSWER:
This should show you how easy it is to add vectors using components. Subtraction is similar except that the components must be subtracted rather than added,
and this makes it important to know whether you are finding or . (Note that .)
Although adding vectors using components is clearly the easier path, you probably have no immediate picture in your mind to go along with this procedure.
Conversely, you probably think of adding vectors in the way we've drawn the figure for Part B.
This justifies the following maxim: Think about vectors geometrically; add vectors using components.
=
,
=
,
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