Physics 2: Quiz 5 Solutions
November 4, 2010
1.
Vector
→
A
has length 4 and a polar angle of 140 degrees, vector
→
B
= 3
ˆ
i
+ 2
ˆ
j
, and vector
→
C
has
length 2 and a polar angle of 30 degrees. Find:
a) the
x
component of
→
C

→
A
We need to start by getting vectors
→
A
and
→
C
into Cartesian coordinates which will allow us to take the
di
ff
erence. We can use the formula:
(
r,
θ
)
→
(
r
cos
θ
, r
sin
θ
)
to convert from polar to Cartesian coordinates.
For
→
A
we have:
→
A
=
(4
,
140
◦
)
=
(4 cos 140
◦
,
4 sin 140
◦
)
=
(

3
.
57
,
2
.
57)
=

3
.
57
ˆ
i
+ 2
.
57
ˆ
j
For
→
C
we have:
→
C
=
(2
,
30
◦
)
=
(2 cos 30
◦
,
2 sin 30
◦
)
=
√
3
,
1
=
√
3
ˆ
i
+
ˆ
j
For the
x
component of the di
ff
erence between the vectors, we just take the di
ff
erence between the
x
components
of each vector:
→
C

→
A
x
=
C
x

A
x
=
√
3

(

3
.
57)
=
5
.
3
b) the magnitude of
2
→
B

3
ˆ
i
To get the magnitude we will need both the
x
and
y
components of the di
ff
erence. We have:
2
→
B

3
ˆ
i
=
2
3
ˆ
i
+ 2
ˆ
j

3
ˆ
i
=
6
ˆ
i
+ 4
ˆ
j

3
ˆ
i
=
3
ˆ
i
+ 4
ˆ
j
so the magnitude is given by:
1
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2
→
B

3
ˆ
i
=
3
ˆ
i
+ 4
ˆ
j
=
3
2
+ 4
2
=
5
c) the polar angle of
→
A

→
B
+ 2
→
C
We can get the polar angle from the Cartesian components. We have:
→
A

→
B
+ 2
→
C
=

3
.
57
ˆ
i
+ 2
.
57
ˆ
j

3
ˆ
i
+ 2
ˆ
j
+ 2
√
3
ˆ
i
+
ˆ
j
=

3
.
57
ˆ
i

3
ˆ
i
+ 2
√
3
ˆ
i
+ 2
.
57
ˆ
j

2
ˆ
j
+ 2
ˆ
j
=

3
.
11
ˆ
i
+ 2
.
57
ˆ
j
And we know that, since the vector is in the second quadrant:
θ
=
arctan
y
x
+ 180
◦
=
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 Fall '10
 ManojKaplinghat
 Physics, Acceleration, Velocity, Elizabeth, polar angle

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