Rotation of a vector and operation we will demonstrate the use of matrix as an operator in this problem. imagine you have a vector r as shown in the figure below. the original vector makes an angle a with x axis. now, we rotate this vector by an angle θ to create our new rotated vector r'. note this operator only rotates the vector but does not change the magnitude of the vector. also note, this example is different from what we discussed before about rotating the coordinate system but leaving the vector unchanged. in the present problem instead we rotate the vector while the coordinate system is unchanged. the original vector r is given by 0 1. assuming, , and y, to be the new coordinates of the rotated vector, express z' and y' in terms of r, α and θ. note rotation does not change the magnitude of the vector but only changos the angle. 2 points 2. now use your rsult and equation 2 to express 2, and y' in terms of z, y and θ. hint: remind yourself about sin a os b + cos asin b. 4 the laws of sin and cos. cos(a+b) cos a cos b-sin asin b and sin(a b) 3. you have now derived r, and y, as a linear combination of and y, hence we should be able to write this in a matrix form as below ru r12 r21 r2 from all the results above, determine matrix r explicitly e. write down all the elements of the matrix r note: it should be only in terms of θ. the physical meaning of r is it is a rotation matrix, when operated on a vector r it rotates the vector by an amount θ and gives you the new components of the new vector r. 4 points . based on the above, now imagine you rotate the vector by an amount -0and we denote this operation as matrix r 1 (calle l the inverse matrix). this is an inverse operation of rotating the vector by same angle θ but in the reverse direction, hence it is the inverse matrix. write down this matrix explicitly. note again it should be in terms of θ only. hint: you should be able to dense this matrz without doing any complicated calculation and directly from the matrir r you hane atready computed 4 points 5. now imagine, you operate this inverse matrix r on your new rotated vector r' to get another new vector r" henoe, the new components of this vector and y" will be given by the following relation. note rijl means it is theリ1h element of the inatrix r and it is not 1/rij. now crnbine equation 3 and equation 4 to note you can relate vector r to vector r. couvince yourself that the following is true let's denote this new matrix r-r as a. you have already determined matrix r- and r above, combine them and do the multiplication of these matrices to determine your new matrix a. write down your a matrix clearly and see if itmakes sense. could you have expected this without any detailed calculation comment on your finding. g points total; 4 points for finding matrir a and 2 points for the last tuo comments