Consider two arbitrary points and We need to find the coordinates of the points and dividing internally and externally respectively, in the ratio .
The approach used in the evaluation of the coordinates of and is analogous to how we derived the section formula in the two dimensional case.
In we have
Assume the coordinates of to be . Thus, the relation in can be written as
Thus, the coordinates of are
The form of the coordinates is the same as in the two dimensional case, as might have been expected. The coordinates of ‘ which divides externally in the ratio can be obtained by substituting for in the coordinates of .
As elementary applications of the section formula, do the following problems :
(a) The mid-points of the sides of a triangle are and . Find its vertices.
(b) Find the coordinates of the centroid of the triangle with vertices
DIRECTION COSINES AND DIRECTION RATIOS
The direction cosines of a (directed) line are the cosines of the angles which the line makes with the positive directions of the coordinate axes.
Consider a line as shown, passing through the origin . Let be inclined at angles to the coordinate axes.
Thus, the direction cosines are given by
Note that for the line (i.e., the directed line segment in the direction opposite to ), the direction cosines will be .
The direction cosines for a directed line not passing through the origin are the same as the direction cosines of the directed line parallel to and passing through the origin.
Note that for any point lying on the line with direction cosines , , such that , the coordinates of will be
The direction cosines of any line will satisfy this relation.
The direction ratios are simply a set of three real numbers , , proportional to , , , i.e.
From this relation, we can write
These relations tell us how to find the direction cosines from direction ratios.
Note that the direction cosines for any line must be unique. However, there are infinitely many sets of direction ratios since direction ratios are just a set of any three numbers proportional to the direction cosines.