 Hi and welcome to the session, I am Shashi and I am going to help you with the following question. Question says, find the coordinates of the point where the line through 516 and 341 crosses the zx plane. First of all let us understand that vector equation of a line which passes through two points whose position vectors are vector a and vector b is r vector is equal to a vector plus lambda multiplied by b vector minus a vector. This is our key idea to solve the given question. Let us now start with the solution. Now we are given that line passes through 516 and 341. Let us represent these points by a and b. So we can write line passes through a having coordinates 516 and point b having coordinates 341. Now a vector represents the position vector of point a so we can write a vector is equal to 5i plus j plus 6k and b vector represents the position vector of point b. So we can write vector b is equal to 3i plus 4j plus k. Now from key idea we know vector equation of a line which passes through two points whose position vectors are vector a and vector v is r vector is equal to a vector plus lambda multiplied by b vector minus a vector. Now we know a vector is equal to 5i plus j plus 6k and b vector is equal to 3i plus 4j plus k. So here in this equation we can write r vector is equal to 5i plus j plus 6k plus lambda multiplied by 3i plus 4j plus k minus a vector that is 5i plus j plus 6k. Now this further implies r vector is equal to 5i plus j plus 6k plus lambda multiplied by minus 2i plus 3j minus 5k. Clearly we can see subtracting these two vectors we get this vector. Now let us assume that point p having coordinates x, 0, z is the point where this line crosses the z-x plane. So we can write let p be the point where this line crosses the z-x plane. Now position vector for point p is xi plus zk. Now let us name this equation as equation 1. Now point p must satisfy this equation as points a, b and p are collinear. We know they lie on the same line. So we can replace vector r by xi plus zk in this equation and we get xi plus zk is equal to 5i plus j plus 6k plus lambda multiplied by minus 2i plus 3j minus 5k. Now equating the coefficients of unit vector i, unit vector j and unit vector k in this equation we get x is equal to 5 minus 2 multiplied by lambda, 0 is equal to 1 plus 3 lambda, z is equal to 6 minus 5 lambda. Now let us name these equations as equation 2, equation 3 and equation 4. Now from equation 3 we get lambda is equal to minus 1 upon 3. Now substituting this value of lambda in equation 2 we get x is equal to 17 upon 3. Substituting lambda is equal to minus 1 upon 3 in equation 4 we get z is equal to 23 upon 3. So we get coordinates of point pr, 17 upon 3, 0, 23 upon 3. So this is our required answer. This completes the session, hope you understood the solution, take care and have a nice day.