Adding and subtracting vectors: The problem with the arrow representation

A small number of studies have investigated student understanding of vector addition and subtraction in generic or introductory physics contexts, but in almost all cases the questions posed were in the vector arrow representation. In a series of experiments involving over 1000 students and several semesters, we investigated student understanding of vector addition and subtraction in both the arrow and algebraic notation (using $\widehat{i}$, $\widehat{j}$, $\widehat{k}$) in generic mathematical and physics contexts. First, we replicated a number of previous findings of student difficulties in the arrow format and discovered several additional difficulties, including the finding that different relative arrow orientations can prompt different solution paths and different kinds of mistakes, which suggests that students need to practice with a variety of relative orientations. Most importantly, we found that average performance in the $ijk$ format was typically excellent and often much better than performance in the arrow format in either the generic or physics contexts. Further, while we find that the arrow format tends to prompt students to a more physically intuitive solution path, we also find that, when prompted, student solutions in the $ijk$ format also display significant physical insights into the problem. We also find a hierarchy in correct answering between the two formats, with correct answering in the $ijk$ format being more fundamental than for the arrow format. Overall, the results suggest that many student difficulties with these simple vector problems lie with the arrow representation itself. For instruction, these results imply that introducing the $ijk$ notation (or some equivalent) with the arrow notation concurrently may be a very useful way to improve student performance as well as help students to learn physics concepts involving vector addition and subtraction.

Figure 1

Examples of one-dimensional questions in arrow and $ijk$ format.

Figure 2

Student performance on one-dimensional vector questions (see Fig. 1). Scores were high for all question types in the $ijk$ format. Scores were lower for subtraction questions, especially for subtraction of opposing vectors ($pmn$). Error bars indicate $\pm 1$ standard error (SE).

Figure 3

Examples of two-dimensional subtraction questions in arrow and $ijk$ format.

Figure 4

Student performance ($N=187$, semester 1) on addition and subtraction of two vectors in the arrow format for questions in which the $x$ components were in opposing directions and the $y$ components were in the same direction (see Fig. 3, left-hand side). The categories represent proportion of students correctly adding (or subtracting) the $x$ components, which are in opposing directions, the $y$ components, which are pointed in the same direction, and the combined total scores of both components. Error bars indicate $\pm 1$ SE.

Figure 5

Student performance ($N=94$, semester 2) on subtraction of two vectors in the arrow format for cases in which either components in both dimensions were in the same direction (aligned) or components in one dimension were aligned and components were opposed in the other dimension (see Fig. 3). For the latter case, the $x$ components were in opposing directions and the $y$ components were in the same direction. The categories represent the proportion of students correctly subtracting the $x$ components (which were opposed or aligned), the $y$ components (which were aligned), and the combined total scores of both components. Error bars indicate $\pm 1$ SE.

Figure 6

Examples of two-dimensional physics context questions in arrow grid, arrow angle, and $ijk$ formats.

Figure 7

Examples of one-dimensional physics context questions in arrow and $ijk$ formats.

Figure 8

Student performance ($N=130$) on two-dimensional (Fig. 6) and one-dimensional (Fig. 7) physics context questions in both the arrow (grid) and $ijk$ formats. Error bars indicate $\pm 1$ SE.

Figure 9

Student performance ($N=221$) on two-dimensional physics context question (Fig. 6) separated by student grades above or below the median course grade (high grade or low grade). Error bars indicate $\pm 1$ SE.

Figure 10

Student performance ($N=221$) on arrow-angle and arrow-grid formats (Fig. 6) with and without first answering a similar $ijk$-format question (“$ijk$ priming”) for low course grade and high course grade students. Error bars indicate $\pm 1$ SE.

Figure 11

Examples of two-dimensional simplified physics context questions in arrow and $ijk$ format.

Figure 12

Assortment of example student solutions to simple two-dimensional physics context questions (Fig. 11) in both arrow and $ijk$ formats. Response rates are in Table 2.

Figure 13

Change in momentum physics context questions in arrow and $ijk$ format.