EEL 3105
‐
Mid Term Exam 1A
‐
Solution
1.
Consider two vectors p and q in two dimensions. Suppose p and q are given in polar coordinates
as
3.
5
4.
8
pr
a
d
qr
a
d
a.
Define z = p+q. Suppose z is expressed as
j
zr
e
. Find r and
. [2pts]
b.
Find the scalar product of p and q. [2pts]
c.
Find the cross product of p and q. [2pts]
Answer:
First, we convert p and q to Cartesian coordinates.
3cos(0.5)
3sin(0.5)
[2.63
1.44]
4cos(0.8)
4sin(0.8)
2.79
2.87
p
q
a.
z=p+q=
5.42
4.31
. We need to express z in polar coordinates to get r and
. We can
calculate r and
to get
22
5.42
4.31
6.92
4.31
arctan
0.67
5.42
r
rads
b.
Scalar product is easy to calculate since we already have the Cartesian representation of
p and q. Thus
.
2.69*2.79
1.44*2.87
11.47
pq
Another way to get this answer is to use the polar representation. Angle between p and
q is [0.8
‐
0.5]rads. Therefore, the scalar product is 12cos(0.3)=11.47.
c.
Cross product of p and q will be a vector perpendicular to the plane of p and q given by
the right hand rule. Let’s represent p and q in the x,y plane via unit vectors along x and y
directions, i, j. Thus,
2.63
1.44
2.79
2.87
2.63*2.87
1.44*2.79
3.54
pij
qij
k
k
where k is the unit vector along the z direction. Another way to get this answer is from
the polar representation for p and q. Again, angle between p and q is 0.3rads.
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View Full DocumentTherefore, the magnitude of pxq is 12sin(0.3)=3.54 and its direction is perpendicular to
the plane of p and q.
2.
Suppose p and q are two vectors in 3 dimensions. Magnitude of p is 3 and the magnitude of q is
5. Moreover, it is known that the magnitude of the scalar product of p and q is equal to the
magnitude of the cross product of p and q.
a.
Find the value(s) of the scalar product of p and q. [2pts]
b.
Suppose we change p and q so that the magnitudes of p and q remain 3 and 5
respectively but the angle between p and q is doubled. What is the scalar product of p
and q? [2pt]
c.
Suppose p and q are changed as described in part b above. What is the magnitude of the
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 Fall '10
 boykins
 scalar product, zw, C. Cross, w. Find r

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