Past exam of the computer science course of the University of Oxford / 2015 / Computer graphics / 2 / b / iii

The multiplication order needs care. Use the standard right-hand rotation matrices, consistent with the printed $G$ and the stated positive $z$ rotation. For column vectors, direct multiplication of the two products in the order printed givesBoth permute the dimensions $(d,w,h)$ into $(h,d,w)$, rather than the required $(w,h,d)$. Therefore, with ordinary column-vector matrix multiplication, neither printed product is the permutation in part (ii). A translation can correct the position of a cuboid, but cannot correct these permuted side lengths for general $w,h,d$. This order mismatch is present in the original printed question.^[ref]

The apparent intended comparison is obtained if “chain” means apply the $y$ rotation first, then the $z$ rotation. Under that interpretation the actual column-vector products have the reverse order:Thus the negative-angle sequence is correct when read as operations performed from left to right. The positive-angle sequence givesIt occupies $[-w,0]\times [0,h]\times [-d,0]$. Apply the world-space translationThis puts the cuboid in $[0,w]\times [0,h]\times [0,d]$ and still maps $z=k\text{step}$ to $y^{\prime }=k\text{step}$. The distinction is purely the action order: ordinary products act rightmost first, while a library may describe an ordered sequence of operations in application order. Giving the coordinate maps makes either convention explicit. If one merely wanted to place the literal printed negative product in the positive octant, its translation would instead be $(h,d,0)$, producing dimensions $(h,d,w)$ and still failing the requested general bounds.