( 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)