 Hi and welcome to the session. Let us discuss the following question. Question says, find the angle between the following pairs of lines. These are the given equations of lines in vector form. First of all, let us understand that if theta is the acute angle between r vector is equal to a1 vector plus lambda multiplied by b1 vector and r vector is equal to a2 vector plus lambda multiplied by b2 vector then cos theta is equal to magnitude of b1 vector dot b2 vector upon magnitude of b1 vector multiplied by magnitude of b2 vector. Now this is the key idea to solve the given question. Let us now start with the solution. Now in the question we are given two equations. They are r vector is equal to 2i minus 5j plus k plus lambda multiplied by 3i plus 2j plus 6k and r vector is equal to 7i minus 6k plus mu multiplied by i plus 2j plus 2k. Now comparing these two equations with the equations in the key idea we get b1 vector is equal to 3i plus 2j plus 6k and b2 vector is equal to i plus 2j plus 2k. Now angle theta between the two lines is given by cos theta is equal to magnitude of b1 vector dot b2 vector upon magnitude of b1 vector multiplied by magnitude of b2 vector. Now substituting corresponding components of b1 vector and b2 vector here we get 3i plus 2j plus 6k multiplied by i plus 2j plus 2k upon Now we know magnitude of b1 vector is given by square root of 3 square plus 2 square plus 6 square. Now magnitude of vector b2 is given by square of 1 plus square of 2 plus square of 2. Now this is further equal to magnitude of 3 plus 4 plus 12 upon square root of 49 multiplied by square root of 9. Clearly we can see 3i multiplied by i is equal to 3 2j multiplied by 2j is equal to 4 and 6k multiplied by 2k is equal to 12. Now here adding these three terms we get 49 so here we can write square root of 49. Similarly here adding these three terms we get 9. So here we can write square root of 9. Now this is further equal to magnitude of 19 upon 7 multiplied by 3. Clearly we can see adding these three terms we get 19. Square root of 49 is 7 and square root of 9 is 3. Now this is further equal to 19 upon 21. So we get cos theta is equal to magnitude of 19 upon 21. So we get value of theta is equal to cos inverse of 19 upon 21. So this is the required value of theta. This completes the session. Hope you understood the solution. Take care. Have a nice day.