1
Vectors
There is a great temptation to put vectors tailtotail when you go to add them.
May all the battles that you wage in your war against that temptation end with
your glorious triumph.
Vectors add headtotail.
We are about to embark on the study of the motion of a particle which does not move only along
a line.
To be sure, we will take things a step at a time and next study the motion of a particle that
stays in a plane, an imaginary flat (but not necessarily horizontal) sheet.
In considering such
motion we find that we can no longer specify the direction of a quantity that has direction by
means of a simple + or – sign as we could in the case of motion along a line.
Now, the direction
of a quantity that has direction, such as the velocity of a car moving along a horizontal surface,
might be specified in words as, for example, “in a compass direction that is 18 degrees north of
due east”.
A number, with or without a plus or minus sign, is no longer sufficient to specify the
magnitude and direction of a velocity, or an acceleration.
Enter the vector.
The vector is a
mathematical entity
1
that has both magnitude and direction.
Mathematicians have devised some operations involving vectors that pertain to physical
quantities that have magnitude and direction.
Knowing about vectors, and the mathematical
operations devised for them, comes in mighty handy in the study of physics.
We start our
discussion of vectors by presenting a couple of different representations of vectors (graphical and
magnitudeanddirection).
Graphical Representation of a Vector
Graphically, a vector is represented by an arrow.
The magnitude of the vector is represented by
the length of the arrow and the direction of the vector is represented by which way the arrow is
pointing.
A vector variable is represented by a letter with
an arrow over it
2
as in the vector
A
v
.
Once we define a vector, we often need to write about the magnitude of that vector—just how
big it is as opposed to both how big and which way it is.
The magnitude of the vector
A
v
can be
written two ways, either
A
v
or
A
.
The first way makes it more obvious that we are dealing with
the magnitude of a vector (rather than some ordinary variable).
The second method is used when
it is already clear from the context that we are dealing with the magnitude of a vector.
You can
never go wrong using
A
v
.
A
is easier to write but the reader might think that it is an ordinary
variable rather than the magnitude of a vector.
1
An entity is just a “something”.
Thus, when we say that a vector is a mathematical entity, all we are saying is that
a vector is a mathematical “something”.
I use the word entity here because one typically uses “a something” when
one is talking about a solid object, and “an entity” when one is talking about a “nonobject” such a thought or a
ghost or, as in the case at hand, a vector.
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 Winter '12
 WilliamPattersonIII
 Trigonometry, triangle, Givens, 108°, 1 37 m

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