 Solution

Take ordinary basal slip in magnesium, $(0001)⟨11\overline{2}0⟩$. In three-index hexagonal notation, $[101]$ denotes a vector proportional to ${\mathbf{a}}_1+\mathbf{c}$, not a cubic Cartesian $(1,0,1)$ with equal axis lengths. The basal slip direction closest to the tensile axis is along ${\mathbf{a}}_1$; its four-index representative is $[2\overline{1}\overline{1}0]$.

For $r=c/a$,Thus the algebraic answer isAn external crystallographic reference gives $a=0.320927\mathrm{nm}$ and $c=0.521033\mathrm{nm}$, hence $r\simeq 1.624$ and ${\sigma }_y\simeq 1.79\mathrm{MPa}$. This is a conditional external-data estimate, not a fabricated value attributed to the supplied Data Book.^[ref]

The tensile axis rotates towards the selected basal slip direction during ideal single slip. As the axis approaches the basal plane, the resolved basal shear eventually becomes small; other modes, hardening and instability can interrupt the single-slip description. TODO: verify the exact magnesium lattice ratio supplied in the historical examination Data Book. Taking a $45^{\circ }$ angle solely from the indices would incorrectly use a cubic metric.