Direct space: points and vectors 

ndimensional Euclidean point space 

ndimensional vector space 
, , 
the field of real numbers, the field of rational numbers, the ring of integers 
L 
lattice in 

line in 
a, b, c; or a_{i} 
basis vectors of the lattice 
a, b, c; or a, b, c 
lengths of basis vectors, lengths of cell edges 

α, β, γ; or α_{j} 
interaxial angles , , 
G, g_{ik} 
fundamental matrix (metric tensor) and its coefficients 
V 
cell volume 
X, Y, Z, P 
points 
r, d, x, v, u 
vectors, position vectors 
r, r 
norm, length of a vector 
x = xa + yb + zc 
vector with coefficients x, y, z 
x, y, z; or x_{i} 
point coordinates expressed in units of a, b, c; coefficients of a vector 

column of point coordinates or vector coefficients 
t 
translation vector 
t_{1}, t_{2}, t_{3}; or t_{i} 
coefficients of translation vector t 

column of coefficients of translation vector t 
O 
origin 
o 
zero vector (all coefficients zero) 
o 
(3 × 1) column of zero coefficients 
a′, b′, c′; or 
new basis vectors after a transformation of the coordinate system (basis transformation) 
r′; or x′; x′, y′, z′; or 
vector and point coordinates after a transformation of the coordinate system (basis transformation) 

column of coordinates after a transformation of the coordinate system (basis transformation) 

image of a point X after the action of a symmetry operation 
; or 
coordinates of an image point 

column of coordinates of an image point 
, or 
(3 + 1) × 1 `augmented' columns of point coordinates or vector coefficients 
