 Good morning friends, I am Purva and today we will work out the following question, find the angle between the vectors i cap minus 2j cap plus 3k cap and 3i cap minus 2j cap plus k cap. Suppose we have two vectors vector A and vector B, then the angle between these two vectors is given by cos theta is equal to vector A dot vector B upon mod of vector A into mod of vector B. So this is the key idea behind our question. Let us now begin with the solution. Now let vector A is equal to i cap minus 2j cap plus 3k cap and vector B is equal to 3i cap minus 2j cap plus k cap. Then we have mod of vector A is given by under root of 1 square plus minus 2 square plus 3 square and we have this is equal to under root of 1 plus 4 plus 9 and this is equal to root 14. So we have got mod of vector A is equal to root 14 and mod of vector B is given by under root of 3 square plus minus 2 square plus 1 square and we have this is equal to under root of 3 square is equal to 9. So we have 9 plus minus 2 square is equal to 4 plus 1 square is equal to 1 and this is equal to root 14. So we have got mod of vector B is equal to root 14. Also we have vector A dot vector B is equal to i cap minus 2j cap plus 3k cap dot 3i cap minus 2j cap plus k cap. Now let we have two vectors vector A which is equal to a1 i cap plus a2 j cap plus a3 k cap and vector B which is equal to b1 i cap plus b2 j cap plus b3 k cap. The dot product of these two vectors is given by vector A dot vector B is equal to a1 into b1 plus a2 into b2 plus a3 into b3. So we will use this formula here to find the value of vector A dot vector B. So we have vector A dot vector B is equal to 1 into 3 plus minus 2 into minus 2 plus 3 into 1 and we get this is equal to 3 plus 4 plus 3 because 1 into 3 gives 3 minus 2 into minus 2 gives 4 and 3 into 1 gives 3 and we get this is equal to 10. So we have got vector A dot vector B is equal to 10. Now we know that angle between two vectors is given by cos theta is equal to vector A dot vector B upon mod of vector A into mod of vector B and we have this is equal to now vector A dot vector B is equal to 10. So we have 10 upon mod of vector A is equal to root 14. So we have root 14 into mod of vector B is also equal to root 14. So we have root 14 and we get this is equal to 10 upon 14 which is equal to canceling the common factor 2 from numerator and denominator. We get 5 in numerator and 7 in denominator. So we get cos theta is equal to 5 upon 7 and we can write this as this implies theta is equal to cos inverse 5 upon 7. Thus we have got our answer as cos inverse 5 upon 7. Hope you have understood the solution. Bye and take care.