An analysis of different representations for vectors and planes in $\mathbb {R}^{3}$

The purpose of this paper is to present an analysis of the difficulties faced by students when working with different representations of vectors, planes and their intersections in \(\mathbb {R}^{3}\). Duval’s theoretical framework on semiotic representations is used to design a set of evaluating activities, and later to analyze student work. The analysis covers three groups of undergraduate students taking introductory courses in linear algebra. Different types of treatments and conversions are required to solve the activities. One important result shows that, once students choose a register to solve a task, they seldom make transformations between different registers, even though this facilitates solving the task at hand. Identifying these difficulties for particular transformations may help teachers design specific activities to promote students cognitive flexibility between representation registers.