Recently, the reviewer [same journal 15 (1978), no. 2, 368--393; MR0483379 (58 \#3385)] introduced a new definition of relative error, called relative precision. Thus if $x$ and $y$ are nonzero real or complex scalars, then (1) $\rho(x, y)=\vert \ln(y/x)\vert $ is the relative precision of $y$ when regarded as an approximation to $x$ (or vice versa). This definition has several advantages over the conventional definition, given by (2) $\rho\sb 0(x, y)=\vert (y-x)/x\vert $; for example, $\rho(x, y)$ is a metric but $\rho\sb 0(x, y)$ is not. \par The present author supplies an elegant generalization of (1) to vectors. If $x$ and $y$ are elements of a real or complex normed space $X$, then (3) $\rho(x, y)=\inf\sb C\int\sb x\sp y\Vert ds\Vert /\Vert s\Vert $, the infimum being taken over all paths $C$ in $X$ linking $x$ and $y$, each norm in (3) being that of $X$. Again, $\rho(x, y)$ is a metric but the conventional measure of relative error, given by (4) $\rho\sb 0(x, y)=\Vert y-x\Vert /\Vert x\Vert $, is not. \par Relations between $\rho(x, y)$, defined by (3), and $\rho\sb 0(x, y)$, defined by (4), are examined by means of several inequalities; in particular it is shown that $\rho\sb 0(x, y)/\rho(x, y)$ tends to unity uniformly on $X$ as $\rho$ or $\rho\sb 0\to0$. Other properties are derived; some are natural analogues of those associated with the scalar definition (1), while others have no scalar analogues. \par The paper concludes with some problems that may need to be solved in subsequent developments of this theory, including an extension to operators, especially matrices. There is also a brief comparison with another measure of relative error for scalars and normed linear spaces that was introduced recently by A. Ziv [Math. Comp. 39 (1982), no. 160, 563--569; MR0669649 (83k:65035)].