 Hello friends, let's discuss the following question. It says, find the vector equation of the line passing through the point 1 to minus 4 and perpendicular to the two lines, x minus 8 upon 3 is equal to y plus 19 upon minus 16 is equal to z minus 10 upon 7 and the line x minus 15 upon 3 is equal to y minus 29 upon 8 is equal to z minus 5 upon minus 5. Let us first know the vector equation of the line passing through the point a1, a2, a3 and parallel to the vector m. Equation of line is given by vector r is equal to vector a plus lambda into vector m, where vector a is the position vector of the point a1, a2, a3. And we also need to know that if we have two vectors b1 and b2, then the vector b1 cross b2 is perpendicular to vector b1 and vector b2. So, this knowledge will work as key idea behind this question. Let us now proceed on with the solution. We are given that the line passes through the point 1 to minus 4. So, here the vector a is 1 i cap plus 2 j cap minus 4 k cap. We are also given that the line is perpendicular to these two lines. So, line is perpendicular to line 1, which is x minus 8 upon 3 is equal to y plus 19 upon minus 16 is equal to z minus 10 upon 7. This is the line 1 and the line is also perpendicular to the line. x minus 15 upon 3 is equal to y minus 29 upon 8 is equal to z minus 5 upon minus 5. Now, line 1 is parallel to the vector 3 i cap minus 16 j cap plus 7 k cap. And similarly, line 2 is parallel to the vector 3 i cap plus 8 j cap minus 5 k cap. Let us call this vector as b1 and this as b2. Now, we know that vector b1 cross b2 is perpendicular to both b1 and b2. So, we first find the vector b1 cross b2, which is the determinant of i cap, j cap k cap when 3 minus 16 7 3 8 minus 5. And it is equal to 80 minus 56 i cap minus minus 15 minus 21 j cap plus 24 plus 48 k cap and it is again equal to 24 i cap plus 36 j cap plus 72 k cap. Now, to obtain the equation of line, we need to have vector a and vector m. We have already obtained vector a and vector m is parallel to the line and also the line is perpendicular to these two lines and also vector b1 cross b2 is perpendicular to the two given lines. That means the required line is parallel to the vector b1 cross b2. So, here the vector m is the vector b1 cross b2. So, the required line is parallel to the vector 24 i cap plus 36 j cap plus 72 k cap plus equation of line passing through the point 1 2 minus 4 and parallel to the vector 24 i cap plus 36 j cap plus 72 k cap is 1 i cap plus 2 j cap minus 4 k cap plus mu into the vector 24 i cap plus 36 j cap plus 72 k cap, which is same as i cap plus 2 j cap minus 4 k cap plus lambda 2 i cap plus 3 j cap plus 6 k cap. We just took 12 common from this and here lambda is 12 mu and this is the required answer and this completes the question. Hope you enjoy this session goodbye and take care.