by Len Cerny

Students have a variety of incompletely formed ideas about basis vectors and components of vectors. Some of this problem stems from vague or varying usage among professional physicists. While professionals almost always know what other professionals mean, students, who have yet to form complete ideas may remain confused.

When students see something like $\Vec{v} = v_x \hat{\mathbf{i}} + v_y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}}$, they often mistakenly assume that the $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ basis vectors have units of length. Some stuudents may assume these basis vectors always have units of length, while other students may assume the basis vectors have whatever units are relevant to the problem. That the scalar proceding the basis vector carries all the units may be something for which students have never been explicitly made aware. Thus students may not distinguish between the dimensionless $\hat{\mathbf{i}}$ and $1\hat{\mathbf{i}}$ in a case where the “1” carries the units.

Another concern is that faculty have different usages of phrases like “the r-component” of a position vector. In some cases faculty will be referring to the radial piece of the position vector, while in other cases they will be referring to the scalar multiplier. Thus for a ring of radius $R$, when saying “r-component” some faculty may mean $R$ while others may mean $R\hat{\mathbf{r}}$. This can become especially problematic when an instructor says something like “the r-component remains constant”, referring to, $R$, the radius of the ring, while a student may interpret it to mean that the radial piece of the vector, $R\hat{\mathbf{r}}$, remains constant. We sometimes use language such as “the radial piece of the vector” or “the magnitude of the radial piece of the vector” to ensure clarity and reduce confusion. However, it may be beneficial to make students directly aware of the ambiguous usage of phrases like “the r-component” and discuss the differences between the meanings.