Solution Let the d . c .' s of the lines be l , m , n . Then l = cos = , m = cos = , n = cos = . Example If a line has direction ratios , – , – , determine its direction cosines.
Solution Direction cosines are ) ) or Example Find the direction cosines of the line passing through the two points (– , , – ) and ( , , ). Solution We know the direction cosines of the line passing through two points P( x , y , z ) and Q( x , y , z ) are given by where PQ = Here P is (– , , – ) and Q is ( , , ). So PQ = ( ( )) ) ( )) Thus, the direction cosines of the line joining two points is Example Find the direction cosines of x , y and z -axis. Solution The x -axis makes angles °, ° and ° respectively with x , y and z -axis.
Therefore, the direction cosines of x -axis are cos °, cos °, cos ° i.e., , , . Similarly, direction cosines of y -axis and z -axis are , , and , , respectively. Example Show that the points A ( , , – ), B ( , – , ) and C ( , , – ) are collinear. Solution Direction ratios of line joining A and B are – , – – , + i.e., – , – , .
The direction ratios of line joining B and C are – , + , – – , i.e., , , – . It is clear that direction ratios of AB and BC are proportional, hence, AB is parallel to BC. But point B is common to both AB and BC. Therefore, A, B, C are collinear points.
EXERCISE . . If a line