### What's Right with $\phat =\phi/|\phi|$ : Student Understanding of Vectors and Scalars

by Len Cerny

#### Overview

When an instructor saw a student with the equation $\phat=\phi/|\phi|$ written on his whiteboard, the instructor commented that they didn't even know where to begin to help the student. Other examples of the procedural and conceptual mathematical errors students make when it comes to vectors include: vector addition, dot products, cross products, and matrix opperations. We have also seen students have trouble confusing vectors and scalars when working with: current, electric potential, position vectors, distances, velocities, and angular velocities. However, while the quantity and variety of errant student conceptions involving scalars and vectors is extensive, students are bringing a host of valuable resources to the table. Recognizing and tapping these resources can be beneficial in helping students construct more usual understandings.

#### Dependable Student Resources

Regarding vectors and scalars, the majority of our incoming juniors in physics can be depended upon to have the ability to:

• Provide examples showing that vectors have direction, while scalars do not
• Break a vector into it's Cartesian components, or find the direction and magnitude of a vector in Cartesian coordinates.
• Apply the equation $\vhat = \vec{v}/|v|$
• Use either $v\cdot w=|v||w|cos \theta$ or $v\cdot w = v_{x}w_{x} + v_{y}w_{y} + v_{z}w_{z}$ or both.

#### Student Confusion and Challenges

We see a variety of student struggles with vectors and scalars during our sequence of five E&M activities, the first of which asks students to find the potential along an axis due to two point charges and the last of which asks students to find the magnetic field in all space due to a spinning ring of charge. We find that several students in the class will have trouble with one or more of the following:

• Thinking of electric potential in terms of vectors. For example, they often claim that there will be a zero electric potential midway between two equally charged points, or in the center of of a charged ring due to “electric forces cancelling.”
• Thinking of electric current in terms of scalars. For example, students will ignore the direction of current when integrating around a ring to find magnetic vector potential. Further discussion of this can be found in the page on student understanding of current.
• The “sea of $r$'s”: Confusing the radius $R$ of a ring, the position vector $\Vec r'$ to a point on the ring, the position vector $\Vec r$ to a point outside the ring, the basis vector $\rhat$, the basis vector $\rhat '$, the radial piece of the position vector to the ring $R\rhat '$, the radial piece of the position vector to the point outside the ring $R\rhat$, and “script $r$ ” = $|\Vec r-\Vec r'|$, which is the distance between a point on the ring and an external point.
• Trying to find $\phat$ using $\phat =\phi/|\phi|$
• Trying to find $\phat$ using incorrect geometry
• Being unsure how to use cylindrical coordinates and rectangular basis vectors to put $|\Vec r-\Vec r'|$ in terms of $\phi$ and known constants

#### Struggling with $\phat$

The direction of current around a ring spinning counterclockwise will always be in the $\phat$ direction. When trying to establish the direction of current, none of our juniors saw it as obvious that in rectangular coordinates $\phat = -\sin{\phi}\, \ii + \cos{\phi}\, \jj$. However, when working in groups of three, most groups were able to establish this relationship within five minutes, especially when given some pointers from the instructor.

In most cases, students drew pictures and tried to establish the relationship using geometry. They often drew a radial vector at angle $\phi$ to the horizontal and then drew a perpendicular $\phat$ vector. However those students that used $\phi = 0, 45^o,$ or $90^o$ often made errors because the picture did not immediately make the errors obvious. For example if $\phi = 0$ were used, students sometimes missed the negative sign needed for the $-\sin{\phi}$ component. On the other hand if $45^o$ were used, students could make errors reversing $\sin{\phi}$ and $\cos{\phi}$. We found it helpful to explicitly tell students that when drawing geometric pictures trying to generalize relationships between angles, that a relatively small angle should be used when drawing an initial angle. In these cases students brought many useful resources to bear on the problem, including understanding the need for a drawing, knowing basic trigonometric relationships, and understanding the geometric relationship between the radial vector $\Vec r$ and $\phat$. Adding the suggestion of drawing a small initial angle added to these students' generally succesful bag of tricks.

There were, however, other students that had farther to go to develop a clear understanding. More than one student confused the radial vector $\rhat$ with the perpendicular vector $\phat$. In these cases, students saw a radial vector as “making an angle $\phi$ with” the horizontal axis and thus confused the direction of the radial vector with the direction of $\phat$. However, even these students knew their basic trig relationships, and when the error was brought to their attention, they could proceed to establish a correct relationship.

There were also at least two students that wrote down $\phat =\phi/|\phi|$. These students were bringing to bear on the problem the relationship $\vhat =\Vec v/|v|$. Since this relationship was very familiar and had never failed them before, they attempted to employ it in this problem. The fact that $\phi$ was not a vector was not immediately obvious to them. In one case a student “solved” the problem, by claiming the radial vector $\Vec r$ was $\phi$ and then solving for $\rhat$ and claiming it was $\phat$. In this case, the student correctly recognized that it was problematic that the angle $\phi$ was not a vector. They were also able to correctly break the chosen vector into components. The students who made the $\phat =\phi/|\phi|$ error had sufficient resources that a few minutes of instructor intervention could allow them to understand why they couldn't generalize $\vhat =\Vec v/|v|$ to this case and also see a valid path for solving the problem.

#### The Sea of "$R$'s"

When trying to find the electric potential in all space due to a ring of charge, students must deal with up to 10 different quantities represented by some form of the letter “R”. In lower-division courses, students rarely dealt with more that two different variables represented by the same letter. In some of these cases student confusion is common, such as understanding the difference between $g$, the local gravitational acceleration and $G$, the universal gravitation constant. Probably the only place where students have already seen so many of the same letter is in momentum problems, which require $x$ and $y$ components of initial and final velocities for two different objects.

In the case of the ring of charge students are trying to wrap their minds around how to deal with the integration while dealing with relatively unfamiliar curvilinear coordinates and also trying to keep track of all the different “$r$'s”. One specific problem that students sometimes encounter is mistaking the position vector $\Vec r'$, from the origin to the ring, for the radius $R$ of the ring. Students correctly notice that the radius of the ring is unchanging but overgeneralize to thinking that $\Vec r'$ also remains constant. Usually students working in groups will have a group member notice the discrepancy and the error is rapidly corrected.

Students also wrestle with the idea that they can use an arbitrary point that is not on the ring as part of creating an integral that can be used to find an electric potential “in all space”. Students often are inclined to make the location of the arbitrary point vary during integration. This is further discussed in variables vs. constants.

One additional thing that has recently come to our attention, is that one student commented that for a long time they were confused by the use of primed variables to refer to anything other than a derivative. We are unsure how common this confusion is.

There is yet another layer of confusion on a more sophisticated level regarding the scalar multiplier combined with the dimesionless basis vectors. Different faculty will use the term “$r$-component” to represent either radial piece of the vector or the scalar portion of the radial piece of the position vector. Students who are unaware of differences in faculty usage may blur these concepts and be confused about the nature of basis vectors. One common misconception amoung juniors is thinking that $\ii$, $\jj$ and $\kk$ have dimensions of length. A more in-depth analysis is available in the discussion of components and basis vectors.

#### Electric Potential: Bringing Vector Resources to the Scalar Table

It is a positive sign that students recognize that there is a connection between electric forces, electric fields, and electric potentials. However, for many students, their high familiarity with vector forces leads them to misapply vector thinking to electric potentials. A significant fraction of our students, perhaps roughly one half, will claim that the electric potential midway between two equal positive charges is zero. When asked to further discribe their thinking, they will frequently mention something about forces cancelling out. This thinking is similarly applied to the center of four point charges or the center of a charged ring. When asked specifically about electric potential being a scalar instead of a vector, students will sometimes claim they already understand this point, but still use various forms of vector addition to result in what they envision is a scalar number.

It should, however, be noted that many of these students also have some ability use valid scalar reasoning in regards to electric fields. Some of the same students who incorrectly claim that there is zero potential midway between two like-charged points will also correctly claim that the there is a zero potential midway between opposite charges.

The vast majority of our incoming juniors have not yet developed a strong intuition for electric potential or any other scalar field, whereas many students do have strong intuitions about sums of forces. The disparity between the strength of these resources should be considered when planning curriculum. We have certainly found no “magic bullet” that solves this problem. We repeatedly use computer visualizations of potential fields, assign a variety of class and homework problems, and work directly with students to help them think in terms of scalar fields. Through this process we find that students leave their junior year with stronger understandings than when they began.

#### Conclusion

Students bring with them a variety of resources from math classes and lower-division physics courses. Many of these resources “transfer” and are applied correctly on demand in upper-division courses. However, there are many cases in which students need additional guidance in order to correctly apply these resources. While nearly all physics juniors can be relied upon to know that a vector has direction while a scalar does not, they may need help in recognizing which quantities are vectors and which are scalars and may also need help understanding the implications of this knowledge in situations such as during integration. Helping students construct a more robust understanding is one of the goals of the “middle division” courses students face as they start their junior year as physics majors.

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