Problem $3$: Determine the indices for the blue and purple vectors shown below.

Algorithm for Miller$-$Bravais Coordinates in HCP:

Determine coordinates of the vector tail and head in terms of three axis $($a$1,$ a$2,$ and z$)$:

a$.$ vector tail: pt$.1$: $x1,y1,$&$z1$

b$.$ vector head, pt$.2$: x$2,$ y$2,$ & z$2$

Tail point coordinates subtracted from head point coordinates and normalized by unit cell dimensions a and $c.$

Adjust to smallest integer values.

Enclose in square brackets, no commas, for three$-$axis coordinates.

Convert to four$-$axis Miller$-$Bravais lattice coordinates using equations below:

$u=\frac{1}{3}(2u^{\text{'}}-v^{\text{'}})v=\frac{1}{3}(2v^{\text{'}}-u^{\text{'}})$

$t=-(u+v)w=w^{\text{'}}$

Adjust to smallest integer values and enclose in brackets $[$uvtw$].$