 Hello and welcome to the session. In this session, first we will discuss direction cosines of a line. Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axis. This directed line L passing through the origin makes angle alpha with the x-axis and will be tau with the y-axis and angle gamma with the z-axis. These angles, alpha, beta, gamma are called direction angles and the cosine of these angles that is cos alpha cos beta cos gamma are called the direction cosines of the directed line L. These direction cosines are denoted by letters L, M and that is we have L equal to cos alpha, M equal to cos beta, N equal to cos gamma. We have a very important relation between the direction cosines of a line given by L square plus N square plus N square is equal to 1. Now we discuss direction ratios of a line. Direction ratios of a line are the numbers which are proportional to the direction cosines of a line. If we have ABC are the direction ratios of a line and MN are direction cosines then since we know that direction ratios are proportional to direction cosines so we have A equal to lambda L, B equal to lambda M and C equal to lambda N where we have lambda is some non-zero real number. When we have the direction ratios given to us and the direction cosines can be obtained as L equal to A upon square root of A square plus B square plus C square, M equal to B upon square root of A square plus B square plus C square and N equal to C upon square root of A square plus B square plus C square. Suppose we are given the direction ratios A, B, C as A equal to 2, B equal to minus 6 and C equal to 3. Now we will find the direction cosines. So direction cosine L is equal to A upon that is 2 upon square root of A square that is 2 square plus B square that is minus 6 square plus C square that is 3 square which is equal to 2 upon 7 then M equal to B upon that is minus 6 upon square root of 2 square plus minus 6 square plus 3 square which is equal to minus 6 upon 7 then direction cosine N equal to C that is 3 upon square root of 2 square plus minus 6 square plus 3 square that is equal to 3 upon 7. So when we are given the direction ratios of a line we can easily find the direction cosines of that line. Next we have direction cosines of a line passing through 2 points consider 2 points P and Q P with coordinates X1, Y1, Z1 and Q with coordinates X2, Y2, Z2 then we have direction cosines of a line joining 2 points P and Q are X2 minus X1 upon PQ, Y2 minus Y1 upon PQ and Z2 minus Z1 upon PQ where PQ is given by square root X2 minus X1 the whole square plus Y2 minus Y1 the whole square plus Z2 minus Z1 the whole square. Consider point A with coordinates 257 point B with coordinates 329 we are supposed to find the direction cosines of a line joining the points A and B. First let's find out what is AB that is equal to square root X2 minus X1 the whole square plus Y2 minus Y1 the whole square plus Z2 minus Z1 the whole square this comes out to be equal to square root 14 that is we have AB is equal to square root 14. Now direction cosines of a line joining the points A and B are given by X2 minus X1 upon AB that is equal to 1 upon square root 14 then next is Y2 minus Y1 upon AB which comes out to be minus 3 upon square root 14 and the next is Z2 minus Z1 upon AB and that is equal to 2 upon square root 14. So 1 upon square root 14 minus 3 upon square root 14 and 2 upon square root 14 are the direction cosines of a line joining the points A and B. This completes the session hope you have understood the direction cosines and direction ratios of a line.