Question

3. (a) (3 marks) If multiplication by matrix A rotates a vector v in the x-y plane through an angle 0, what is the effect of multiplying v by A?. (b) (3 marks) Describe the geometric effect of multiplying a vector x by the following matrix (cos0 – sin? 0 -2 sin 0 cos 0 ) | 2 sin 6 cos sin0 – cosa e ) (c) (4 marks) In three dimensional space, find a matrix that rotates a vector by 30° coun- terclockwise about the z-axis (that is looking down on x-y plane) and then by 45° coun- terclockwise about the x-axis (that is looking down on the y-z plane).

Answer 1

plane a) Given tasks are ; Multiply rector with matrix A in any Through an angle o. 1 Effect of multiplying by AT Answers: 3) let's asume vector vewy) = ] auestion it is 3 7 V (1,2) 40-anis In the ginen question it is not mentioned whether to et rotate clockwise cor) t Anti-clockwise. I la lets rotate the vector clock wis. T matrix for cloche wire rotation & A caso sine et assume - sino caso Multiply rector with V → A. V ool 2 3 Fig (1) o = A R 340098 [2le Vector rotated clockwise Now THY to the . (1,2) i clockwise ") Apply traus pose taken matrix

AT multiply 2, with 2009[-][-]°63-030 +17 0 so it is clear that Multiplication of Transpose of a Matrix with vector always. (2) 2 VCt, 2) gore will be opposite to PR AE to work wise The moltiplication of a Matrix hat-3-2-o va 3 Shaxis to vlas, (2,7) [ vAZv. AT Ginen matrix Costosinto 2 sino Coso 7 esino coso sin costo s o let's assume os T/4 and Nector x = [! > Col T74)-8in?(1/4) - : -2sixma) Cos (17) 2 80€774) ((1/4) BirdEW/4) Cos?W4) Ch 2.1 .ro A.x lo-17 Effect?? Multiplication of vector se with Matrix A GO) willa always rotate towards "conter clock wiser

by 300 canter about the i find Matrix that sotates vector close through (2-axis) Cay plane) ii) And then rotante canterclock wire (x-axis) (y-2) plane} In 2-0 place for clockwise sotation Answers ue wire matrix. 1 347 clock wise rcoso sing sino Care 232 Sati-clockwise start costs tate Coso sino Ustne ose In the same way for our understanding with no rotation matrix sril Hanis yaxis z-axis. It was mentioned to rotate vector by 30° conter clock wise about 2 axis to keeping 2-axis remain unchanged. x, yaxis changed froso-sino o sino Coso o Let's assume rector = Gren o=30°. D Lo 0

R & R₂0V T2 to] 0.22 20-016 2.73 Ito w Lo or 13 4t Ril I, I figure (1 72 figure (2) 3o the matrix that solates a vector by 3 conter clockwise about the a raxis is cose - sino ol I sino caso ol 1 x Bol ond 1 Matrix that states conter clock wire by 45° about the x-axis is S 0 0 Jo coro -sino | O -KI 8 ne exe O= 0 yote ||2.79-01212 Lotta [²] 13 4.9