 Hello and welcome to the session. I am Deepika here. Let's discuss a question which says show that the line joining the origin to the point 211 is perpendicular to the line determined by the points 3 5 minus 1 and 4 3 minus 1. Now we know that how to find the angle between two lines. Let the direction cosines of the two lines L1 and L2B, L1, M1, N1 and L2, M2, N2 respectively and theta is the angle between them then cos theta is equal to mod of L1 L2 plus M1 M2 plus N1 N2. Again these two lines are perpendicular theta is equal to 90 degree. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now we will first find out the direction cosines for the line joining the origin to the point 211. So the direction cosines for the line joining origin the point with coordinate 211 are 2 minus 0 upon under root of 2 minus 0 whole square plus 1 minus 0 whole square plus 1 minus 0 whole square and 1 minus 0 upon under root of 2 minus 0 whole square plus 1 minus 0 whole square plus 1 minus 0 whole square and 1 minus 0 upon under root of 2 minus 0 whole square plus 1 minus 0 whole square plus 1 minus 0 whole square or 2 upon under root of 2 square that is 4 plus 1 square that is 1 plus 1 and 1 over under root of 2 square 4 plus 1 plus 1 and 1 over under root of 4 plus 1 plus 1 or 2 over under root of 6 1 over under root 6 and 1 over under root 6. So these are the direction cosines of the line joining the origin to the point whose coordinates are 211. Now we will find out the direction cosines of the line joining the points 35 minus 1 and 43 minus 1. So the direction cosines for the line joining the points with coordinates 35 minus 1 and 43 minus 1 are 4 minus 3 over under root of 4 minus 3 whole square plus 3 minus 5 whole square plus minus 1 minus minus 1 whole square that is minus 1 plus 1 whole square and 3 minus 5 over under root of 4 minus 3 whole square plus 3 minus 5 whole square plus minus 1 plus 1 whole square and minus 1 plus 1 upon under root of 4 minus 3 whole square plus 3 minus 5 whole square plus minus 1 plus 1 whole square or 1 over under root of 1 plus 4 and minus 2 over 1 plus 4 and 0 over under root of 1 plus 4 which is 0 only or 1 over under root 5 and minus 2 over under root 5 and 0. So these are the direction cosines of the line joining the points with coordinates 35 minus 1 and 43 minus 1 like theta is the angle between lines L1 and L2 then cos theta is equal to mod of 2 over root 6 into 1 over root 5 plus 1 over root 6 into minus 2 over root 5 plus 1 over root 6 into 0 because according to a key idea if the direction cosines of the two lines L1 and L2 be L1 M1 M1 and L2 M2 M2 respectively then cos theta is equal to mod of L1 L2 plus M1 M2 plus M1 M2 and this is equal to mod of 2 over root 30 minus 2 over root 30 plus 0 and this is again equal to 0 so cos theta is equal to 0 implies theta is equal to 90 degree thus the given two lines are perpendicular to each other. So this completes our session. I hope the solution is clear to you. Bye and take care.