 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that, find the coordinates of the foot of the perpendicular from P with coordinates 2, 5, 1 on the line joining A with coordinates 2, 4, 2 and B with coordinates 0, 7, 9. We know that if two lines are perpendicular to each other then A1, A2 plus B1, B2 plus C1, C2 is equal to 0 where A1, B1, C1 and A2, B2, C2 are called direction ratios of the two lines. Direction ratios of any line say PQ are given by x2 minus x1, y2 minus y1, z2 minus z1 where x1, y1, z1 and x2, y2, z2 are the coordinates of point P and Q respectively. With this key idea let us proceed with the solution. Let L be the foot of the perpendicular from point P to AB and let L divides AB in the ratio is to 1 that is L divides AB in the ratio K is to 1 and we need to find the coordinates of point L. Now we can find coordinates of point L using the section formula which states that the x coordinate is given by m1 x2 plus m2 x1 whole upon m1 plus m2. The y coordinate is given by m1 y2 plus m2 y1 whole upon m1 plus m2 and the z coordinate is given by m1 z2 plus m2 z1 whole upon m1 plus m2 which divides the join of the points x1, y1, z1 and x2, y2, z2 in the ratio m1 is to m2. Since L divides AB in the ratio K is to 1 therefore coordinates of point L are given by the x coordinate is given by m1 x2 that is 0 into K which is 0 plus m2 x1 that is 1 into 2 which is 2 upon m1 plus m2 that is K plus 1. The y coordinate is given by m1 y2 that is 7 into K plus m2 y1 that is 1 into 4 which is 4 upon m1 plus m2 that is K plus 1 and the z coordinate is given by m1 z2 that is 9 into K 9 K plus m2 z1 that is 1 into 2 which is 2 upon m1 plus m2 that is K plus 1 or the coordinates of L can be written as 2 upon K plus 1, 7 K plus 4 whole upon K plus 1, 9 K plus 2 whole upon K plus 1. Next we shall find direction ratios of line PL and AB using the key idea which states that direction ratios of any line say PQ are given by x2 minus x1, y2 minus y1, z2 minus z1 where x1, y1, z1 and x2, y2, z2 are the coordinates of point P and Q respectively. Therefore, direction ratios of perpendicular PL are given by x2 minus x1 that is 2 upon K plus 1 minus 2 y2 minus y1 that is 7 K plus 4 upon K plus 1 minus 5 z2 minus z1 that is 9 K plus 2 upon K plus 1 minus 1 which is equal to 2 minus 2 K minus 2 upon K plus 1, 7 K plus 4 minus 5 K minus 5 upon K plus 1, 9 K plus 2 minus K minus 1 upon K plus 1 which is equal to minus 2 K upon K plus 1, 2 K minus 1 upon K plus 1, 8 K plus 1 upon K plus 1. Also, direction ratios of line AB are given by x2 minus x1 that is 0 minus 2, y2 minus y1 that is 7 minus 4, z2 minus z1 that is 9 minus 2 which is equal to minus 2, 3, 7. Since PL is perpendicular to AB and we know that if two lines are perpendicular to each other then a1, a2 plus b1, b2 plus c1, c2 is equal to 0 where a1, b1, c1 and a2, b2, c2 are the direction ratios of the two lines. Since PL is perpendicular to AB, we have a1, a2 plus b1, b2 plus c1, c2 is equal to 0 where a1, b1, c1 and a2, b2, c2 are the direction ratios of line AB and PL respectively which implies a1, a2 that is minus 2 into minus 2 K upon K plus 1 plus c1, c2 that is 7 into 8 K plus 1 upon K plus 1 is equal to 0 which implies 4 K plus 6 K minus 3 plus 56 K plus 7 whole upon K plus 1 is equal to 0. On solving further, we get 66 K plus 4 is equal to 0 which implies K is equal to minus 2 upon 33. We know the coordinates of point L in terms of K so on putting the value of K in the expression we get the coordinates of L that is the X coordinate is given by 2 upon K plus 1 that is 2 upon K plus 1 2 upon minus 2 by 33 plus 1 the Y coordinate is given by 7 K plus 4 upon K plus 1 that is 7 into minus 2 upon 33 plus 4 upon K plus 1 that is minus 2 by 33 plus 1 the Z coordinate is given by 9 K plus 2 upon K plus 1 that is 9 into minus 2 by 33 plus 2 upon minus 2 by 33 plus 1 which is equal to 2 upon 31 into 33, 118 upon 33 into 33 upon 31 which implies 48 upon 33 into 33 upon 31 which gives the coordinate of point L as 66 upon 31, 118 upon 31, 48 upon 31. Hence the coordinates of the foot of the perpendicular is given by 66 upon 31, 100 18 upon 31, 48 upon 31 which is the required answer. This completes our session. Hope you enjoyed this session.