Question

Exercise 9.49 system is defined by the three points A, B, and C. The i planes A through B. The ż axis is defined by the cross product of AR e right-handed, barred xyz system is def the plane ABC. The X axis runs from 9.20 The right-hand defined by the plane ABC axis lies on axis lies on the same side of the i axis as point C. into AC, so that the + ansformation matrix [R] relating the two coordinate systems, ie soll = 12 -13], find the components of Find the orthogonal transformation If a vector in the barred system is {vt - in the unbarted system c. (3.9.-2) B (4,6,5) 2,3)

Answer 1

Aust stepO; a first define the position vectors of Points A, B, and c., VANB and re XYZ Coordinates the the in A = \f+ 3 + 3+ VB = 41+ 6î ts 8c = 3î gî - 2 These with are Points the xtz A, B, C. coordinates system step Hext Calculate a vector, AB, that points along the x-axis from Point A to Point B, By subtracting &A and re. - AB=8B-JA = (u 1 + 65+5#- (11 +2 +3A) = 3 Î +uſ teh step of Now Calculate a vector Ac, those fonts from Point A to Point B by subtracting &A from &

Ас - та - XA - (31+af - »8) - (of tzņ +38) + 9 + 4 1 - step Produce a vedor Pointing along the Zlaxis by talking the Cross Product of AB and Ac. Z' = AB XAL = (3 t'+49 +24) * (2 s^+79-51) +{}= =)-4)3] [cv(a)-6)(-5) + [(s) () =(4)() A = -34 +145 +13 À Step(b 1 Produce a vector Pointing along the I axis, y' by taking the Cross Pooduct of 2' and AB y' = 2'x AB = (-34f +19 9 +15F) x (zî +49 428) - [CM)) = (13) () * +{0.00+(3)e + [(-34)_w)(19) c) = -49 +1673" -LASK

that points along the vector from A to B, stept Create the unit vector the x' aris i , by dividing AB , by its mag nitude. Y = AB LABI O co +uft98). (3.744$ 728) =0.57 +0.74 289 +0.3714€ step3 1 - Creak the unit vector that points along with y'aris Points along the ŷ , by dividing the vector that y'ar's y', by its magnitude. :-14I+ 10.75 -1938 Jen464 1035-1934) 4 M310739 1938) = -0.06337701483459 – 0.8128

stept Create the unit vector that Doints along the z' axis. Å by dividing te vector that points along the zlaxis . z' and dividing it by its magnitude. ² = 2 1741 -34 +19 f +13 h (-34ğ tla9 H131?). (-34+10° +138) = -0.8280 € 70.4627 +0.32667 of compute the values of elements of the Cosine matrix. Q. from the basis vectors, I, JIK, in, and a A1 A1 A11 IT J. i kiil 11 A 1 10.557 -0.0633 |- 6.8280 0.7428 0.1839 0.4627 6:3714 -0.8728 0.3266 " The direction of relating the two cosive matrix [a] Coorihale bases 105,

0.587 Git 428 0.4839 0.3714) -0,8728 -0.06 32 0.8280 0.4627 10.3266 6 Delie te vector v', in the Pointed System '=- +3† The tours Pose of to cosine matrix, a', is needed to exproces te vector qu te unprivad basis, This is simply the previous cesine of matrix with the sove and columus swapped. 10.557 -0.6633 at - 0.7428 0.4839 0.3714 -0.8728 -0.8280 -0.8728 03266 Finaly , to get te vector in the unfoined fove u, multiply to Privad vector u, by the tous fase of the Cosine mail orx QT. onte left. Us Tv (0.557 -6.0633 -0.8282 0.7428 0.4834 -0.8727 10.3714 -0.8728 013266 = - 113041 +2:3909+2:565ť