Question

Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0) = sin() cos(0) Consider now the unit vector v = [1,0)" in a two dimensional plane. Compute R(O)v. Repeat your computations this time using w = [0, 1]". What do you observe? Try thinking in terms of pictures, look at the pair of vectors before and after the action of R(O). 23. You may have recognised the two vectors in the previous question to be the familar basis vectors for a two dimensional space, i.e., i and j. We can express any vector as a linear combination of j and j, that is, u= ai + bi for some numbers a and b. Given what you learnt from the previous question, what do you think will be result of R(0) u? Your answer can be given in simple geometrical terms (e-g., in pictures). 24. Give reasons why you expect R(4+0) = R(0) R(). Hence deduce that cos(8+) = cos(8) cos($) - sin() sin() sin(( + 0) = sin(0) cos(m) + sin(o) cos(0) 25. Give reasons why you expect R(0) R(0) = R(0) R(O). Hence prove that the rota- tion matrices R(0) and Ro) commute. 26. Show that det(R(0)) = +1. 27. Given the above form for R(0) write down, without doing any computations, the inverse of R(O).

Answer 1

R(0) rotates the vector u to angle a anticlook wine 22) R(0) = r conca) -sinla). Sinco) Cosco) R(2) V = ſcoocm) -Sim Co) sinca coco) No R(Q) W = cos(o) - sin(a) Sinco) +.2012) V² COD0 sino 0][-]-[ ][i]. [1 sina രായ 19 (cosa, Sina) (-sino copy ab 142 (10) Thamrin R(@) rotates rector to every angle a anti clockwine. R()u za.coop te osima) i + (asinet.co.za); 23) w = [0] = ait by U= Cai+b;) ton a a ROU. cona sino Sina Cood a cosa - osina asino tocasa

of oping the operation of rotating 24) R(a+) denotes notation ano rector angle af and R (2) R ($) a rector v an-angle of and then to an angle so it is same as R (a + c@+6 R (2) R (f)v By . RC°)•(P(4)) 27 het va vas V RC0708) - Sincato) T (8+of) (02 cos Card) sin card) cos(at & 1[!] (1) sin (018) ER (@ RC) RCB]V = R(S)[R(8)x] Rear icon of cond -sing ind cool R(0) cool sind 310 sin (02 cono asin sind Cona sino -6112 Sino c 0090 6096 - sin@sind cond +copa sing $. R[@tf)= R(O)R(4) from C1 and (1) CO9 (044) = conoconf - siasing Sin (014) = sinacond + cosa sing

25 Rotating a rector. to angle o at first and then to an angle & is given by RODRC). ie vis rolated to angle auf altogether. Again . R.(O)R(O) v. ib rotating & and then to o. ie angle at & anticheck- v to angle -wire altogether. So BLOOC8) = R(8) RCB) R[o+). RO) R14) Cosa-sina 1) coop - -sing sina cosa sinh cosa cosa cond -sinasing - cosasing - sina соор sina cool + cosa sing singsina + cosa oss too long sin(a+8 sincato) Co2 (9++) " R(6) R (a)= T cood -sinf 0000 Sina Sino Cona sing con lo sino cosa t sina cos o losa-sinasin coolsina - sin & cosa coops tooo como -smo sino coo (@+9) - SYNC@ 7.6) sin card) con cato) 26) clet (Pc)) - Copa [ . -sina cosa y sina sina Cona -

to an V 27) R(0) sistates a vector v angle o anticlock - cuire The inverse of this operation would get the vector v back to its original position This is possi by rotating v So inverse of R (2) would be R(-2) con (-a) - Sin (-2) Sin (-Q) possible by -O angle. [RCO)]" cos (-a) = cona t sino -sino cona