a family of directions
In a highly symmetric crystal, several different directions are really the same to the atoms — a physicist would measure identical properties along each. A family of directions collects all such symmetry-equivalent arrows under one name, written in angle brackets, <uvw>. The single symbol <100> stands for the whole set of directions that the crystal's symmetry can turn one into another. It is the difference between naming one street and naming every street that runs the same way in a perfectly gridded city.
How many members a family has depends entirely on the crystal system, because it is the symmetry that decides which arrows are interchangeable. In a cubic crystal, where the three axes are identical, <100> contains six directions — the three cube edges and their reverses [100], [010], [001], [-100], [0-10], [00-1]. The family <110> contains the twelve face diagonals, and <111> contains the eight body diagonals. You generate the members by permuting the numbers and flipping their signs in every way the symmetry allows.
Because slip in metals happens along the most densely packed directions, and those densely packed directions form a symmetry family, engineers talk about slip along a whole family at once — for example slip along <110> directions in face-centred cubic metals. That single family symbol quietly stands for all twelve equivalent glide directions available to a dislocation.
In a cubic crystal <110> means twelve directions at once: [110], [101], [011], [1-10], [10-1], [01-1] and their six reverses. In a tetragonal crystal, where c differs from a and b, that same set splits — [110] type stays a family but [101] and [011] become a separate one, because the symmetry no longer treats c like a.
Angle brackets gather all symmetry-equivalent directions; the count depends on the crystal system.
The familiar cubic counts (six for <100>, twelve for <110>, eight for <111>) do not carry over to lower-symmetry systems. Always ask which symmetry operations actually apply before listing members.