 Hi friends, I am Purva and today we will work out the following question. Find the angle between the following pairs of lines. Vector r is equal to 3i cap plus j cap minus 2k cap plus lambda into i cap minus j cap minus 2k cap and vector r is equal to 2i cap minus j cap minus 56k cap plus mu into 3i cap minus 5j cap minus 4k cap. Now if theta is the acute angle between the following pairs of lines, that is vector r is equal to vector a1 plus lambda into vector b1 and vector r is equal to vector a2 plus mu into vector b2. Then cos theta is given by cos theta is equal to mod of vector b1 dot vector b2 upon mod of vector b1 into mod of vector b2. So this is the key idea behind our question. Let us begin with the solution now. Now we are given the lines as vector r is equal to 3i cap plus j cap minus 2k cap plus lambda into i cap minus j cap minus 2k cap and vector r is equal to 2i cap minus j cap minus 56k cap plus mu into 3i cap minus 5j cap minus 4k cap. Now by comparing these two equations with these two equations in the key idea we can clearly see that here vector b1 is equal to i cap minus j cap minus 2k cap and vector b2 is equal to 3i cap minus 5j cap minus 4k cap. Now vector b1 dot vector b2 is equal to vector b1 is equal to i cap minus j cap minus 2k cap dot vector b2 is equal to 3i cap minus 5j cap minus 4k cap and this is equal to now 1 into 3 is 3 and i cap dot i cap is equal to 1 we know that minus 1 into minus 5 is plus 5 again j cap dot j cap is equal to 1 and minus 2 into minus 4 is plus 8 again k cap dot k cap is equal to 1. So we get this is equal to 16. Now mod of vector b1 is equal to under root of 1 square plus minus 1 whole square plus minus 2 whole square and this is equal to under root of 1 plus 1 plus 4 which is equal to under root 6 and mod of vector b2 is equal to under root of 3 square plus minus 5 whole square plus minus 4 whole square and this is equal to under root of 9 plus 25 plus 16 and this is equal to under root 50. So mod of vector b1 into mod of vector b2 is equal to under root 6 into under root 50 which is equal to under root 300. Now by key idea we know that the angle between two lines is given by cos theta is equal to mod of vector b1 dot vector b2 upon mod of vector b1 into mod of vector b2. Putting the values we get that is cos theta is equal to mod of 16 upon under root 300 which is further equal to 16 upon 10 root 3 and this implies cos theta is equal to 8 upon 5 root 3 which implies theta is equal to cos inverse 8 upon 5 root 3. Thus we have got our answer as theta is equal to cos inverse 8 upon 5 root 3. Hope you have understood the solution. Bye and take care.