 Hello and welcome to the session. In this session we are going to discuss the following question which says that find the sign of the angle between the vectors i cap plus 2 j cap plus 2 k cap and 3 i cap plus 2 j cap plus 6 k cap We know that sign of angle theta is given by modulus of vector a into vector b by modulus of vector a into modulus of vector b. With this key idea, let us proceed with the solution. Let vector a be given by i cap plus 2 j cap plus 2 k cap and vector b be given by 3 i cap plus 2 j cap plus 6 k cap then vector a cross vector b is given by we have written it in the determinant form in which the first row consists of the unit vectors i cap, j cap, k cap The second row consists of the coefficient of the unit vectors of vector a that is 1, 2 and 2 and the third row consists of the coefficient of the unit vectors of vector b that is 3, 2 and 6 Now solving it further we get i cap into 20 minus 4 minus j cap into 6 minus 6 plus j cap into 2 minus 6 which is equal to 8 i cap minus 0 j cap minus 4 k cap Now modulus of the cross vector vector a and b is given by square root of 8 square plus minus 4 square which is equal to square root of 64 plus 16 That is square root of 80 which is equal to 4 into square root of 5. Now modulus of vector a is given by square root of 1 square plus 2 square plus 2 square which is equal to square root of 9 that is 3 And modulus of vector b is given by square root of 3 square plus 2 square plus 6 square which is equal to square root of 9 plus 4 plus 36 that is square root of 49 which is given by 7 And we need to find the sign of the angle between the two vectors which is given by sin theta is equal to modulus of vector a into vector b upon modulus of vector a into modulus of vector b Modulus of vector a cross b is given by square root of 5 Modulus of vector a is 3 and modulus of vector b is 7 And we have sin of angle theta is equal to modulus of cross vector vector a and vector b upon modulus of vector a into modulus of vector b which is equal to 4 into square root of 5 upon 3 into 7 That is 4 square root of 5 by 21 Therefore, sin of the angle between the two vectors that is i cap plus 2 j cap plus 2 k cap and 3 i cap plus 2 j cap plus 6 k cap is given by sin to square root of 5 by 21 which is the required answer. This completes our session. Hope you enjoyed this session.