Linear Algebra

140 Chapter Two. Vector Spacestake as a unit the length of each square’s side. That is, we naturally adopt thisbasis.( ) ( )3.34 0〈 , 〉0 3.34Then we can describe, say, the corner in the upper right of the picture above as3⃗β 1 + 2⃗β 2 .Another crystal from everyday experience is pencil lead. It is graphite,formed from carbon atoms arranged in this shape.This is a single plane of graphite, called graphene. A piece of graphite consistsof millions of these planes layered in a stack. The chemical bonds between theplanes are much weaker than the bonds inside the planes, which explains whypencils write — the graphite can be sheared so that the planes slide off and areleft on the paper.We can get a convenient unit of length by decomposing the hexagonal ringinto three regions that are rotations of this unit cell.The vectors that form the sides of that unit cell make a convenient basis. Thedistance along the bottom and slant is 1.42 Ångstroms, so this( ) ( )1.42 1.23〈 , 〉0 .71is a good basis.Another familiar crystal formed from carbon is diamond. Like table salt itis built from cubes but the structure inside each cube is more complicated. Inaddition to carbons at each corner,there are carbons in the middle of each face.