矢豫分析:矢量之间运算的定义 点乘的意义: 绝对长度(模):A.A=∑A1A1=AF 14/384 单位矢豫 A为1,分豫为方向角 A. B 夹角: COsA·B=0÷A⊥B AB A 方向角:CB4 分豫A;是模的投影 ose
14/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ ✿➛✛➾➶➭ ýé⑧Ý (✜): A~ · A~ = X 3 i=1 AiAi = |A~ | 2 ü➔➙þ➭ A~ |A~ | ✜➃1, ➞þ➃➄➉✍ ❨✍➭ A~ · B~ |A~ ||B~ | ≡ cos θ A~ · B~ = 0 ⇐⇒ A~ ⊥B~ ✟✟✟ ✟✟✟ ✟✟✟✯ θ ✲ B~ A~ ➄➉✍➭ cos θi ≡ Ai |A~ | ➞þAi➫✜✛Ý❑
矢豫分析:矢量之间运算的定义 叉乘:(叉积,矢积,外积) 基矢的叉乘: 15/384 e 0 e1×e ∈ike 2,3的偶置换例:(123,(231,(312 2,3的奇置换例:∈132,(213,321 其它 例 e111,∈223,313 ose
15/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ ✄➛➭ (✄➮➜➙➮➜✠➮) ➘➙✛✄➛➭ ~e1 × ~e1 = ~e2 × ~e2 = ~e3 × ~e3 = 0 ~e1 × ~e2 = −~e2 × ~e1 = ~e3 ~e2 × ~e3 = −~e3 × ~e2 = ~e1 ~e3 × ~e1 = −~e1 × ~e3 = ~e2 ➼ ~ei × ~ej = X 3 k=1 ijk~ek ijk ≡ 1 1, 2, 3✛ó➌❺ ⑦➭ 123, 231, 312 −1 1, 2, 3✛Û➌❺ ⑦➭ 132, 213, 321 0 Ù➜ ⑦➭ 111, 223, 313
矢均分析:矢量之间运算的定义 A×B=(A161+A22+A363)×(B1+B262+B36) 16/384 A1B1e1×e1+A1B2e1×e2+A1B3e1×e +A2B1e2×e1+A2B2e2×e2+A2B3e2×e3 +A3B1e3×e1+A3B2e3×2+A3B33×e3 (A1B2-A2B1)e3+(A3B1-A1B3)e2+(A2B3-A3B2)e1 B×A=∑A1B1问 ij, k=1 ∈k具有如下注质 uk交换任两指标一次变号fuk=-6fik=-6ki=-n ∑mk=0n1m-mn ose
16/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ A~ × B~ = (A1~e1 + A2~e2 + A3~e3) × (B1~e1 + B2~e2 + B3~e3) = A1B1~e1 × ~e1 + A1B2~e1 × ~e2 + A1B3~e1 × ~e3 +A2B1~e2 × ~e1 + A2B2~e2 × ~e2 + A2B3~e2 × ~e3 +A3B1~e3 × ~e1 + A3B2~e3 × ~e2 + A3B3~e3 × ~e3 = (A1B2 − A2B1)~e3 + (A3B1 − A1B3)~e2 + (A2B3 − A3B2)~e1 = −B~ × A~ = X 3 i,j,k=1 ijkAiBj~ek ijkä❦❳❡✺➓➭ ijk✂❺❄ü➁■➌❣❈Ò ijk = −jik = −kji = −ikj X 3 k=1 ijklmk = δilδjm − δimδjl
矢册分析:矢量之间运算的定义 u=(n-05n) 17/384 ∑=∑6n2-6)=9-3=6 j, k=1 A×B的计算过程可简写为: AxB=∑Ax∑B=∑AB问x∑AB A×A=01 A×B).C=(∑(uABA)∑Cn∑aABC2Ek:n i、j,k=1 n=1 i、j,k,n=1 ∑ ik A, B; CnOKn-∑ABC1行列式;循环 ij, k, n=l ose
17/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ X 3 j,k=1 ijkljk = X 3 j=1 (δilδjj − δijδjl) = 3δil − δil = 2δil X 3 i,j,k=1 ijkijk = X 3 i,j=1 (δiiδjj − δijδji) = 9 − 3 = 6 A~ × B~ ✛❖➂▲➜➀④✕➃➭ A~ × B~ = X 3 i=1 Ai~ei × X 3 j=1 Bj~ej = X 3 i,j=1 AiBj~ei × ~ej= X 3 i,j,k=1 AiBjijk~ek A~ × A~ = 0 (A~ × B~ ) · C~ = ( X 3 i,j,k=1 ijkAiBj~ek) · X 3 n=1 Cn~en = X 3 i,j,k,n=1 ijkAiBjCn~ek · ~en = X 3 i,j,k,n=1 ijkAiBjCnδkn = X 3 i,j,k=1 ijkAiBjCk ✶✎➟;❒❶
矢量分析:矢量之间运算的定义 叉乘的意义: 绝对长投(模):|A×B|=|A|B|sinb 18/384 方向:A×B⊥A,BA‖B←→A×B=0 A×B A×B2=(A×B)·(A×B)}=(A.A)(B.B)-(A.B)(A·B A21B|2-(A|B|cos6)2=|A|21B2(1-cos2) AB A×B)·A=0(A×B)·B ose
18/384 JJ II J I Back Close ➙þ➞Û: ➙þ❷♠✩➂✛➼➶ ✄➛✛➾➶➭ ýé⑧Ý(✜): |A~ × B~ | = |A~ ||B~ |sin θ ➄➉➭ A~ × B~ ⊥A~ , B~ A~ k B~ ⇐⇒ A~ × B~ = 0 ✘✘✘✘✘✘✿ PPPPPPPPPq A~ × B~ ✻ θ A~ B~ |A~ × B~ | 2 = (A~ × B~ ) · (A~ × B~ ) = (A~ · A~ )(B~ · B~ ) − (A~ · B~ )(A~ · B~ ) = |A~ | 2 |B~ | 2 − (|A~ ||B~ | cos θ) 2 = |A~ | 2 |B~ | 2 (1 − cos2 θ) = |A~ | 2 |B~ | 2 sin2 θ (A~ × B~ ) · A~ = 0 (A~ × B~ ) · B~ = 0