上浒文通大学 Vector manipulations SHANGHAI JIAO TONG UNIVERSITY Vector addition: A=Aeio B B=Bei08 C=A+B =B+A C=A+B=Aeio+Beioa B -(AcOS0+Bcose)+ (a) i(Asin0,+Bsing) A=C-B Vector substraction: B A=C-B=Ceioc-Beios B -(Ccosec -Bcoseg)+ (b) j(Csinec-Bsineg) Figure (a)Vector addition and(b)vector subtraction
Vector manipulations jθ Vector addition: θ A A j A e θ A = B j B e θ B = θ B =( cos cos )+ A B j j Ae Be A B θ θ θ θ =+= + + CAB =( cos cos )+ j( sin sin ) A B A B A B A B θ θ θ θ + + θ θ C C B j j Ce Be θ θ A CB =−= − Vector substraction: θ B =( cos cos )+ C B Ce Be C B θ θ =−= − − A CB j( sin sin ) C B θ C B − θ Figure (a) Vector addition and ( b) vector subtraction
图上游充通大学 SHANGHAI JIAO TONG UNIVERSITY y y本 A C C B B A B (a) (b) (a) (b) Figure Graphical solution of case 2:(a)given C,A,and B; Figure Graphical solution of case 3:(a)given C,A,and B; (b)solution for A,B and A',B*. (b)solution for A and B. N√ 0√√o Vy OOv C=A+B C=A+B
Figure Graphical solution of case 2: (a) given C, Aˆ , and B; (b) solution for A, Bˆ and A’, Bˆ’. Figure Graphical solution of case 3: (a) given C, Aˆ, and Bˆ; (b) solution for A and B. CAB = + √ √ O √ √ O CAB = + √ √ O √ O √
上泽充通大粤 SHANGHAI JIAO TONG UNIVERSITY Vector rotation: R=Reio /R' R'-Re1o=R ej(0+o) R
Vector rotation: Y j R e θ R = j j( ) θ R R ' ( ) 'e j j R e ϕ θ +ϕ R R= = O R θ ϕ X
上浒文通大学 SHANGHAI JIAO TONG UNIVERSITY @=i2+(-1)+k4(rad/s) Vector cross product: r=i(-1)+j10+k2(mm), i方kijk A=iA,+jA,+kA. D=ω×r=AA,A=2-14 B=iB,+jB,+kB BB,B-1102 =i(-2-40)+j(-4-4)+k(20-1) =i(-42)+j(-8)+k(19)mm/s i k AxB= A A B. By B. AxB=i(A,B.-A.B)+j(A.B,-A,B.)+k(A B,-A,B A×B=ABsin0
ω = i2 + j(−1) + k4 (rad /s). r = i(−1) + j10 + k2 (mm), Vector cross product: Ax Ay Az 2 1 4 i j k i j k υ = ω× r = = − Ax Ay Az A = i + j + k Bx By Bz ( 2 40) ( 4 4) (20 1) 1 10 2 = i − − + j − − + k − − Bx By kBz B = i + j + = i(−42) + j(−8) + k(19)mm /s i j k r v A×B = Ax Ay Az ω Bx By Bz ( ) ( ) ( ) × = AyBz − AzBy + AzBx − AxBz + k AxBy − AyBx A B i j A B× = ABsinθ
上游充通大学 SHANGHAI JIAO TONG UNIVERSITY ixj=kj×i=-k ixk=i k×i=-i k×i=jixk=-j i×i=i×ji=k×k=0 j×i=-k ×i=
i × j = k j × i = −k j × k = i k × j = −i k × i = j i × k = − j i × i = j × j = k × k = 0 j j j j