a negative sign is to be taken for l , m and n . For any line, if a , b , c are direction ratios of a line, then ka , kb , kc ; k ≠ is also a set of direction ratios. So, any two sets of direction ratios of a line are also proportional. Also, for any line there are infinitely many sets of direction ratios.
. . Direction cosines of a line passing through two points Since one and only one line passes through two given points, we can determine the direction cosines of a line passing through the given points P( x , y , z ) and Q( x , y , z ) as follows (Fig . (a)).
Fig . Let l , m , n be the direction cosines of the line PQ and let it makes angles α , β and γ with the x , y and z -axis, respectively. Draw perpendiculars from P and Q to XY-plane to meet at R and S. Draw a perpendicular from P to QS to meet at N.
Now, in right angle triangle PNQ, ∠ PQN= γ (Fig . (b). Therefore, cos γ = NQ Similarly cos α = and cos β= Hence, the direction cosines of the line segment joining the points P( x , y , z ) and Q( x , y , z ) are where PQ = A Note The direction ratios of the line segment joining P( x , y , z ) and Q( x , y , z ) may be taken as x – x , y – y , z – z or x – x , y – y , z – z Example If a line makes angle °, ° and ° with the positive direction of x , y and z -axis respectively, find its direction cosines.