Section 7: Properties of Adding Vectors
Identities and Inverses
Do you remember the property of having an identity element from the Trip Line
and Road Coloring Units? What would the identity element be for vector addition?
If there is an identity element, then we can also ask if every vector has an inverse
vector that gets us back to the identity element. Remember, the identity element
of an operation is the element that “doesn’t change things.” In the relay race,
what would have to happen in a relay race leg for no change to happen?
Team Work
As a team, discuss what the identity element for vector addition would be. In the
context of the relay race, what would this mean? Write your answer below and
present your ideas to the class.
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What were your conclusions concerning the identity element? Remember, an
identity element is always the element that doesn’t change things. For the relay
race, what would “not change things” mean? If the first leg of a relay race is as
pictured below, what would the second leg be, if it did not change anything?
Start
v1
The second leg would have to have no displacement, for it not to change anything. (In a real relay race, this would mean the person would have to fall down
and not move at all. In our type of relay race, the person would have to stack 0
cubes.). This reasoning is very similar to what you experienced with the Trip Line.
Now that we understand what the identity element is for vectors, what icon should
we use to represent it? Is this identity element a vector?
Team Work
As a team, create an icon for the identity element for vector addition. Do you
think this identity element is a vector? Why or why not?
Class Work
Collect all of team icons together on chart paper. As a class, choose which one of
the icons the class will use when it writes mathematical sentences that include this
identity element.
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Again, just as we did in the Road Coloring unit and the Trip Line unit, now that
we understand the idea of identity element, we will try to understand the idea of
an inverse in the Relay Race.
Team Work
As a team, what would be the inverse of a vector in the relay race? Write your answer below and present your ideas to the class. Also, decide on an icon or symbolic
representation for the inverse of a vector, and present this icon to the class.
Class Work
Collect all of team icons together on chart paper. As a class, choose which one of
the icons the class will use when it writes mathematical sentences about inverses.
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The key is to understanding the inverse of a vector (or any mathematical object) is
that the inverse turns a vector into the identity element. is the vector that has zero
displacement. In other words, after the first two legs, there would have to be zero
displacement. Well, if the second leg was exactly the same length but in the opposite direction, then we would be back at the start line.
Start
v1
v2
Remember, as long as we don’t change their length or direction, we can pickup
vectors and move them anywhere. What would the inverse look like if we moved
it to the start line?
Start
v1
v2
Therefore, the inverse of the vector v1 is the vector that is exactly as long as v1, but
goes in the opposite direction.
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Individual Work
For each of the diagrams below, draw the inverse of the given vector. Make sure
the vector you draw begins at the start line. Label the vector you draw as “additive
inverse of v1”
Example:
Start
v1
Additive inverse of v1
Exercises:
Start
a.)
v1
Start
b.)
v1
Start
c.)
v1
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What symbols did your class decide to use for the identity element of vector addition? Whatever that symbol was you can continue to use it whenever you write
things in class. But since this material was written before you made that choice,
we will have to use a symbol that may be different.
What was the class decision about this “no movement” identity element?. Is it a
vector? Mathematicians decided it was best to think of this “no movement” as a
vector as well. Since “no movement” seems a lot like the number 0, we will start
to use the symbol 0 for this identity element vector, writing it in bold italics.
Also, if we have a vector v, we will use the symbol -v for the additive inverse of v.
Once again this should remind you of symbols that we used in both the Trip Line
and the Road Coloring unit.
What are the properties we have verified so far?
1.)
Commutative Property
For any two vectors u and w, we have that
2.)
u+w=w+u
Associative Property
For any three vectors u , v, and w, we have that
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(u + v) + w = u + (v + w)
3.)
Existence of an Identity Element
There exists a vector 0, so that for any other vector v
4.)
v+0 =0+v=v
Existence of an Additive Inverse
For any vector v, there exists an additive inverse vector -v, so that
v + (-v) = (-v) + v = 0
Team Work
As a team, take each of the properties above and translate them into your own
words. Write a sentence that describes, in your own words, what each property is
trying to say. Then come up with an example in the context of the relay race that
shows what you mean by your sentence. Report to the class your examples
and sentences.
1.) Commutative property
For any two displacement vectors u and w, we have that u + v = v + u.
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2.) Associative Property
For any three vectors u, v, and w, we have that
(u + v) + w = u + (v + w).
3.) Existence of an Identity Element
There exists a vector 0, so that for any other vector v,
v + 0 = 0 + v = v.
4.) Existence of an Additive Inverse.
For any vector v, there exists an additive inverse vector -v, so that
v + (-v) = (-v) + v = 0.
Class Work
As a class, put the each of the properties on a sheet of chart paper and list below
them the “translations” and examples from the teams.
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