12
Cartesian Representation
A vector can be broken up into the sum of two vectors, one parallel to the xaxis, one parallel to the y
axis.
The scalar lengths of the two vectors above are given from trigonometry, in equation 35 of text. These
are called the components of the vector, and can be positive, negative, or zero.
a
x
=
r
a
cos
θ
and
a
y
=
r
a
sin
(
Polar to Cartesian
conversion)
The Cartesian representation of a two dimensional vector is the two components, (a
x
, a
y
)
It is more common in physics and engineering to write the Cartesian form with unit vectors,
r
a
=
a
x
ˆ
i
+
a
y
ˆ
j
The unit vectors have magnitude 1.
Cartesian to Polar Conversion
. If we know the components we get the polar form from
r
a
=+
a
x
2
+
a
y
2
and
tan
=
a
y
a
x
. There is one remaining wrinkle: inverse tangent is a function
that returns two distinct answers, the (angle) and the (angle + pi). Calculators return only one of these, it
is up to you to determine the proper quadrant and pick the proper angle.
e.g.
r
a
=−
3.0
cm
()
ˆ
i
+−
6.0
cm
ˆ
j
r
a
=
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 Fall '09
 LIND
 Vectors, Vector Space, Dot Product, unit vectors

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