 Hello and welcome to the session. In this session, first we will discuss equation of a line through a given point and parallel to a given vector. Consider a point A with position vector A with respect to the origin of the rectangular coordinate system. Let L be the line which passes through this point A and this line is parallel to a given vector B. Let R be the position vector of a point P on the line. Then we have vector equation of a line that passes through the given point whose position vector is vector A parallel to a given vector B is given by vector R equal to vector A plus lambda into vector B. And if we have the coordinates of the point A as x1, y1, z1 that is coordinates of this point A are given by x1, y1, z1 and ABC are the direction ratios of the line. Then equation of the line through a point A and having ABC as direction ratios is given by x minus x1 upon A equal to y minus y1 upon V equal to z minus z1 upon C. This is the Cartesian equation of the line and if we are given the direction cosines that is L, M, N of the line then we say equation of the line is given by x minus x1 upon L equal to y minus y1 upon M equal to z minus z1 upon N. Also if we are given vector B as I cap plus B j cap plus C k cap then we say ABC are direction ratios of the line. Let's find out the vector equation of a line that passes through a point with position vector given by vector x equal to 2 I cap minus j cap plus 4 k cap and is parallel to the vector given by vector y I cap plus j cap minus 2 k cap. Now the vector equation of the line is given by vector r equal to vector x plus lambda into vector y where lambda is some non-zero real number that is vector r equal to 2 I cap minus j cap plus 4 k cap plus lambda into I cap plus j cap minus 2 k cap. This is the required vector equation of the line. Next we discuss equation of a line passing through two given points. Consider two points A and B lying on the line. The point A has coordinates x1, y1, z1 point B has coordinates x2, y2, z2. We have a point P also on the line with coordinates x, y, z position vector of point P with respect to the origin is given by vector r, position vector of point A is given by vector A and position vector of point B is vector B. So we have vector equation of a line which passes through two points whose position vectors are vector A and vector B is given by vector r equal to vector A plus lambda into vector B minus vector A. We have the coordinates of point A as x1, y1, z1 and coordinates of point B as x2, y2, z2 and the coordinates of point P are given by x, y, z. So Cartesian equation of a line that passes through two points A with coordinates x1, y1, z1 and B with coordinates x2, y2, z2 is given by x minus x1 upon x2 minus x1 equal to y minus y1 upon y2 minus y1 equal to z minus z1 upon z2 minus z1. Let's try and find out the Cartesian equation of the line passing through the points A with coordinates 2 minus 1, 4 and point B with coordinates 1, 1 minus 2. From the point A we have x1 equal to 2, y1 equal to minus 1 and z1 equal to 4. From point B we have x2 equal to 1, y2 equal to 1 and z2 equal to minus 2. So Cartesian equation of the line passing through points A and B is given by x minus x1 that is x minus 2 upon x2 minus x1 that is 1 minus 2 equal to y minus y1 that is y minus minus 1 upon y2 minus y1 that is 1 minus minus 1 equal to z minus z1 that is z minus 4 upon z2 minus z1 that is minus 2 minus 4. So this comes out to be x minus 2 upon minus 1 equal to y plus 1 upon 2 equal to z minus 4 upon minus 6. This is the required Cartesian equation of the line. This completes the session. Hope you have understood how we find the equation of a line through a given point in parallel to a given vector and equation of a line passing through two given points.