 Good morning friends. I am Purva and today we will discuss the following question. Find the vector equation of the plane passing through the intersection of the planes vector r dot 2i cap plus 2j cap minus 3k cap is equal to 7 and vector r dot 2i cap plus 5j cap plus 3k cap is equal to 9 and through the point 2 comma 1 comma 3. Let pi 1 and pi 2 be two planes with equations vector r dot n1 cap is equal to d1 and vector r dot n2 cap is equal to d2 respectively. The position vector of any point on the line of intersection must satisfy both the equations. Now if vector t is the position vector of a point on the line then for all real values of lambda we have vector t dot n1 cap plus lambda into n2 cap is equal to d1 plus lambda d2 Now since vector t is arbitrary it satisfies for any point on the line hence the equation vector r dot vector n1 plus lambda into vector n2 is equal to d1 plus lambda d2 represents a plane pi 3 which is such that if any vector r satisfies both the equations pi 1 and pi 2 it also satisfies the equation pi 3. Hence the equation of the plane passing through intersection of the planes vector r dot vector n1 is equal to d1 and vector r dot vector n2 is equal to d2 is given by vector r dot vector n1 plus lambda into vector n2 is equal to d1 plus lambda d2 So this is the key idea behind the question. Let us begin with the solution now. Now we have given the planes as vector r dot 2i cap plus 2j cap minus 3k cap is equal to 2i cap plus 2j cap and vector r dot 2i cap plus 5j cap plus 3k cap is equal to 9. Now the position vectors of the planes are vector n1 is equal to 2i cap plus 2j cap minus 3k cap d1 is equal to 7 by comparing this equation we can clearly see that here vector n1 is equal to 2i cap plus 2j cap minus 3k cap and d1 is equal to 7. Again by comparing this equation with this equation we can clearly see that vector n2 is equal to 2i cap plus 5j cap plus 3k cap and d2 is equal to 9. Now by key idea we know that the equation of the plane passing through intersection of the planes is given by vector r dot vector n1 plus lambda into vector n2 is equal to d1 plus lambda d2. Now putting the values of vector n1 vector n2 d1 and d2 in the above relation we get vector r dot now vector n1 is equal to 2i cap plus 2j cap minus 3k cap plus lambda into vector n2 which is equal to 2i cap plus 5j cap plus 3k cap is equal to d1 which is equal to 7 plus lambda into d2 and d2 is equal to 9 we can write this as or vector r dot 2 plus 2 lambda into i cap plus 2 plus 5 lambda into j cap plus minus 3 plus 3 lambda into k cap is equal to 7 plus 9 lambda. We mark this as equation 1 and here we have lambda is some real number. Now taking vector r is equal to xi cap plus yj cap plus zk cap and putting in this equation 1 we get xi cap plus yj cap plus zk cap dot 2 plus 2 lambda into i cap plus 2 plus 5 lambda into j cap plus minus 3 plus 3 lambda into k cap is equal to 7 plus 9 lambda. Or we can write this as 2 plus 2 lambda into x plus 2 plus 5 lambda into y plus minus 3 plus 3 lambda into z is equal to 7 plus 9 lambda. Or we can write this as 2x plus 2y minus 3z plus 2x plus 5y plus 3z into lambda is equal to 7 plus 9 lambda. We mark this as equation 2. Now since the plane passes through the point 2 comma 1 comma 3 it must satisfy equation 2. So putting x is equal to 2, y is equal to 1 and z is equal to 3 in this equation 2 we get 2 into 2 plus 2 into 1 minus 3 plus 2 into 2 plus 5 into 1 plus 3 into 3 into lambda is equal to 7 plus 9 lambda. This implies 4 plus 2 minus 9 plus 4 plus 5 plus 9 into lambda is equal to 7 plus 9 lambda. This implies minus 3 plus 18 lambda is equal to 7 plus 9 lambda which further implies minus 3 minus 7 is equal to 9 lambda minus 18 lambda which further implies minus 10 is equal to minus 9 lambda and this implies lambda is equal to 10 upon 9. Now putting this value of lambda in equation 1 we get vector r dot 2 plus 2 into lambda and lambda is equal to 10 upon 9 i cap plus 2 plus 5 into 10 upon 9 into j cap plus minus 3 plus 3 into 10 upon 9 into k cap is equal to 7 plus 9 into 10 upon 9 we can write this as or vector r dot 38 upon 9 i cap plus 68 upon 9 j cap plus 3 upon 9 k cap is equal to 153 upon 9 or we can write this as vector r dot 38 i cap plus 68 j cap plus 3 k cap is equal to 153. Thus we have got the equation of the planas vector r dot 38 i cap plus 68 j cap plus 3 k cap is equal to 153. This is our answer. Hope you have understood the solution. Bye and take care.