given straight lines ( ) ( ) The parametric form of vector equations of the given straight lines are r = ( ) ( ) and r = ( ) ( ) Comparing the given two equations with tb sd we have , , , k b k c k d Clearly, b is a scalar multiple of d , and hence the two straight lines are parallel. We know that the shortest distance between two parallel straight lines is given by | ( Now, ( Therefore, d = | | | | Example . Find the coordinate of the foot of the perpendicular drawn from the point ( , , ) to the straight line ) ( . Also, find the shortest distance from the given point to the straight line.
Comparing the given equation ) ( with tb , we get , and . We denote the given point ( , , ) by D , and the point ( , , ) on the straight line by F . If F is the foot of the perpendicular from D to the straight line, then F is of the form ( , , ) ( ) ( ) DF OF OD tk Since b is perpendicular to DF , we have b DF = ( ) ( ) ( ) ⇒ ⇒ Therefore, the coordinate of F is ( , , ) Now, the perpendicular distance from the given point to the given line is DF = ( ) DF = + − units. Fig.
. F Line D Vector - - Applications of Vector Algebra