If you gave me a particular vector
A
as an arrow of some length
A
and orientation
θ
relative to the
x
-axis, what do I use for
A
x
and
A
y
? You
can see from trigonometry that
A
x
=
A
cos
θ
(2.6)
A
y
=
A
sin
θ
.
(2.7)
Conversely, given the components, the length and angle are
A
=
A
2
x
+
A
2
y
(2.8)
θ
=
tan
−
1
A
y
A
x
.
(2.9)
Eqns. 2.6 to Eqn. 2.9 will be invoked often. So please commit them to
memory.

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Motion in Higher Dimensions
21
If you give me a pair of numbers, (
A
x
,
A
y
), that’s as good as giving
me this arrow, because I can find the length of the arrow by Pythago-
ras’ theorem and I can find the orientation from tan
θ
=
A
y
A
x
. You have the
option of either working with the two components of
A
or with the arrow.
In practice, most of the time we work with these two numbers, (
A
x
,
A
y
).
In particular, if we are describing a particle whose location is the
position
vector
r
, then we write it in terms of its components as
r
=
i
x
+
j
y
.
(2.10)
The
changes
in
r
are the
displacement vectors
and examples are
A
and
B
in
Figure 2.2 that described the two hikes.
I have not given you any other example of vectors besides the dis-
placement vector, but at the moment, we’ll define a vector to be any object
that looks like some multiple of
i
plus some multiple of
j
. If I tell you to add
two vectors
A
and
B
, you have got two options. You can draw the arrow
corresponding to
A
and attach to its end an arrow corresponding to
B
,
and then add them, as in Figure 2.2. But you can also do the bookkeeping
without drawing any pictures as follows:
A
+
B
=
i
A
x
+
j
A
y
+
i
B
x
+
j
B
y
(2.11)
=
i
(
A
x
+
B
x
)
+
j
(
A
y
+
B
y
)
(2.12)
so that the sum
C
is the vector with components (
A
x
+
B
x
,
A
y
+
B
y
).
In the above, I have used the fact that vectors can be added in any
order. So I grouped the things involving just
i
and likewise
j
. Then I argued
that since
i
A
x
and
i
B
x
are vectors along
i
, their sum is a vector of length
A
x
+
B
x
also along
i
. I did the same for
j
.
In summary if
A
+
B
=
C
(2.13)
then
C
x
=
A
x
+
B
x
(2.14)
C
y
=
A
y
+
B
y
(2.15)
which can be summarized as follows:

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22
Motion in Higher Dimensions
To add two vectors, add their respective components.
An important result is that
A
=
B
is possible only if
A
x
=
B
x
and
A
y
=
B
y
. You cannot have two vectors equal without having exactly the same
x
component and exactly the same
y
component. If two arrows are equal,
one cannot be longer in the
x
direction and correspondingly shorter in
the
y
direction. Everything has to match completely. The vector equation
A
=
B
is actually a shorthand for two equations:
A
x
=
B
x
and
A
y
=
B
y
.
2.4
Choice of axes and basis vectors
I have in mind a vector whose components are 3 and 5. Can you draw
the vector for me? If you immediately said, “It is 3
i
+
5
j
,” you’re making
the assumption that I am writing the vector in terms of
i
and
j
. I agree
i
and
j
point along two natural directions. For most of us, given that the
blackboard or notebook is oriented this way, it is very natural to line up
our axes with it. But there is no reason why somebody else couldn’t come
along and say, “I want to use a different set of axes. The

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- Spring '17
- Jeremiah Badana
- The Bible, The Land