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Student difficulties with graphical representation of vector products: crossing and dotting beyond ts and is*
Warren M. Christensen, Ngoc-Loan Nguyen, and David E. Meltzer
Iowa State University
*Supported in part by NSF REC #0206683

One of the questions administered to the students in the Spring 221 class was given to the
Summer 221 and 222 students as a question on an exam. Due to the constraints of the exam
we were forced to condense the responses from 10 down to 5. The question for the 222
class was put into the context of a charged particle in a magnetic field.

In an effort to test students understanding of the graphical representation of scalar and vector products, a
four-question quiz was administered to students in a first-semester calculus-based physics course [221]
during the spring and summer of 2004, as well to as students in a second semester calculus-based physics
course [222] during the summer of 2004. The questions and results are below. (Questions were
administered during the final week of the spring course, and near the mid-point of the summer courses.)

Multiple choice options for Spring 221

1i 2i 1i 2i cos i

1i 2i 1i 2i cos i

Since all vectors are of equal length :
X i cos i

Since all vectors are of equal length :
X i cos i ;

cos A 0, cos B 0, cos C 0

cos C cos B cos A

X A 0, X B 0, X C 0

XC XB XA

Correct Responses
N
% of N

Correct Responses
N
% of N

Students failing to recognize XA is smallest (i.e.,
responding with answers A, B, C, E , F, or G):
N

% of N

221 Spring

168

28%

221 Summer

36

222 Summer

41

Multiple choice options for Summer 221/222

221 Spring

168

68%

221 Spring

168

52%

221 Summer

36

64%

221 Summer

36

58%

222 Summer

41

76%

222 Summer

41

61%

Students failing to recognize XC is the greatest (i.e.,
responding with answers A, B, C, D, E, or F):
N

% of N

221 Spring

168

27%

22%

221 Summer

36

20%

222 Summer

41

Students failing to recognize XC is zero (i.e.,
responding with answers A, C, D, E, F, H, or I):
N

% of N

221 Spring

168

28%

22%

221 Summer

36

17%

222 Summer

41

Those students who appeared to utilize a component method for calculating the scalar products were
successful in obtaining a correct answer. Students often abandoned a component method in favor of some
equation representation [i.e., |1A||2A|cos()], with varying degrees of success.

Spring 221 (N = 168)
A
B
C
D
E
F
G
H
18% 0% 40% 1% 6% 4% 17% 1%

Students failing to recognize XA is negative (i.e.,
responding with answers A, B, C, D, or E):
N

% of N

221 Spring

168

32%

17%

221 Summer

36

33%

20%

222 Summer

41

27%

Typical student response when failing to recognize XA is negative (seen in 221 and 222 students):

I
3%

J
5%

In order to get down to five choices, we removed B, D, E, F,
and H. Even though choices E and F had more responses
than choice I, studies have shown that some students have
difficulty distinguishing the direction of a vector from that
of a vector in the opposite direction (Nguyen and Meltzer,
2003). The substantial number of students selecting
response G seems to support that notion. Therefore, we
retained response I as a choice for the summer exam
question, renaming it response C.

I know C has to be 0, because cos(90) = 0, and you use the absolute values so [the magnitudes]
must be >0. The angle isn't negative because it's the angle between the two vectors.

Summer 221 (N = 48)
A
B
C
D
E
23% 50% 4%
6% 17%

Summer 222 (N = 56)
A
B
C
D
E
25% 68% 4%
4%
0%

One sixth (17%) of 221 students responded that the vector product has
a magnitude of zero.
On Question 3, 15% of 222 students had explicitly given zero for the
magnitude of the vector product of two perpendicular vectors (i.e.,
stated that XC = 0 on that question). On this exam question, by contrast,
none gave that response. It is possible that the magnetic-field context of
the 222 exam question was responsible for this difference.
Both 221 and 222 students seem to have significant difficulty in
applying the right-hand rule, as ~25% of both classes chose the
direction opposite to the correct response on the exam question. This is
consistent with the responses to Question 4.

Many students chose to be the tip-to-tail angle, without recognizing the need to use parallel
vector transport.

The biased nature of a random sample
when using an online medium
In the process of testing students understanding of vector and scalar products, we were
offered an opportunity to use an online medium, WebCT, to administer a quiz. Complying
with the instructors request, we divided our six question quiz into two 3-question quizzes.
At the end of the semester, we analyzed the overall class scores (final numerical grade) of
every student in the class. Below is the score distribution for the two groups that took
quizzes (combined) and the one that did not.
Course Scores
20%
18%
16%
14%
Percentage of Students

1i 2i 1i 2i sin i
Since all vectors are of equal length :
X i sin i
sin C sin A sin B
XC XA XB

12%
Quiz 1 and 2

10%

No Quiz

8%
6%
4%

Correct Responses
N
% of N

Students failing to recognize XC is the greatest (i.e.,
responding with answers A, B, C, D, E, or F):
N
% of N

2%

Correct Responses
N
% of N

221 Spring

206

58%

221 Summer

36

50%

221 Spring

206

58%

222 Summer

41

56%

221 Summer

34

53%

222 Summer

41

61%

Students failing to recognize XB is smallest (i.e.,
responding with answers A, B, E, F, H, or I):
N
% of N

221 Spring

206

36%

221 Spring

206

35%

221 Summer

36

42%

221 Summer

36

42%

222 Summer

41

37%

222 Summer

41

39%

Typical student response for an incorrect calculation of the magnitude of the vector product:

0%
30-35

Several students attempted to use a matrix method to calculate the cross product but there were no apparent successes.

40-45

45-50

50-55

55-60

60-65

65-70

70-75

75-80

80-85

85-90

90-95

95-100

Course Scores

Statistical analysis shows the following:
Students responding with answer F (the
directions of the vector products are reversed):
N

% of N

221 Spring

206

0%

221 Summer

34

222 Summer

41

Students responding with answer E (all vector
products are pointing out of the page):
N

% of N

221 Spring

206

16%

22%

221 Summer

34

11%

20%

222 Summer

41

5%

Because for cross prod[uct] it is (1)(2)cos and you can factor out the (1)(2)
Many students used a similar cos reasoning; they not only failed to recognize XC as being the greatest quantity,
but most often determined that it was zero.

35-40

The absence of F responses in the spring 221 class is rather troublesome. Before the quiz was administered we
speculated that F would be the most common incorrect answer. Our expectations were confirmed during the summer
classes for both 221 and 222, but the absence of such responses in the spring 221 class is unexplained.
None of the students who selected response E provided an explanation.

Descriptives
SCORE

No Quiz
Quiz 1
Quiz 2

N
293
167
204

Mean
63.8
71.3
70.9

Std. Error
.687
.844
.818

95% Confidence Interval for
Mean
Lower Bound
Upper Bound
62.4
65.1
69.6
72.9
69.3
72.6

The mean course score for students who took Quiz 1 (71.3) is statistically identical to the
score of those who took Quiz 2 (70.9), but significantly larger (p < 0.0001) than that of those who took no quiz (63.8) [a difference equivalent to one full letter grade].

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