( ii ) The coordinates of any point , which divides the join of points P ( x1 , y1 , z1 ) and Q ( x2 , y2 , z2 ) in the ratio m : n externally are
( mx2 – nx1 / m – n , my2 – ny1 / m – n , mz2 – nz1 / m – n )
( iii ) The coordinates of mid-point of P and Q are ( x1 + x2 / 2 , y1 + y2 / 2 , z1 + z2 / 2 )
( iv ) Coordinates of the centroid of a triangle formed with vertices P ( x1 , y1 , z1 ) and Q ( x2 , y2 , z2 ) and R ( x3 , y3 , z3 ) are ( x1 + x2 + x3 / 3 , y1 + y2 + y3 / 3 , z1 + z2 + z3 / 3 )
( v ) Centroid of a Tetrahedron
If ( x1 , y1 , z1 ), ( x2 , y2 , z2 ), ( x3 , y3 , z3 ) and ( x4 , y4 , z4 ) are the vertices of a tetrahedron , then its centroid G is given by
( x1 + x2 + x3 + x4 / 4 , y1 + y2 + y3 + y4 / 4 , z1 + z2 + z3 + z4 / 4 )
Direction Cosines
If a directed line segment OP makes angle α , β and γ with OX , OY and OZ respectively , then Cos α , cos β and cos γ are called direction cosines of up and it is represented by l , m , n .
i . e ., l = cos α m = cos β and n = cos γ
If OP = r , then coordinates of OP are ( lr , mr , nr )