a. b-(a,, b,e )=a,b, e, e) =a, oy=a,b axb=a; e,xb,e, =e, xe, a, b,=Eika, 6,ek 2 Ena1a=a×a=0 G×b)a=a1(e) Eiia, b, c,e k a.b.c C.a..bC,= b2
1 2 3 1 2 3 1 2 3 b b b a a a e e e a b a e b e e e a b a b e i i j j i j i j ijk i j k = = = = ijka j ak = a a = 0 ( ) ( ) ( ) ( ) ijk i j k l l ijk i j l k l a b c a b e c e a b c e e = = 1 2 3 1 2 3 1 2 3 c c c b b b a a a a b c a b c = ijk i j l kl = ijk i j k = ( ) ( ) ( ) i i j j i j i j ai bj ij ai bi a b = a e b e = a b e e = =
例题1.展开下列求和式,(1)a1a;(2)a 解:(1)aa1=a1a1+a2a2+a13a3 (2) ajk l kaki t aiai arra =a1C1+a1221+a1331+a2112+a2a22+a23432 +a31a13+a32a23+a343
例题1. 展开下列求和式， 解： 1 1 2 2 3 3 (1).ai ai = a a + a a + a a 3 1 1 3 3 2 2 3 3 3 3 3 1 1 1 1 1 2 2 1 1 3 3 1 2 1 1 2 2 2 2 2 2 3 3 2 1 1 2 2 3 3 (2). a a a a a a a a a a a a a a a a a a a a a a a a a a j k kj k k k k k k + + + = + + + + + = + + (1). ; (2). . ai ai a jkakj
例题2已知,下=+2元-5,币=37-1+k,求 (1)·;(2)v×w (4)l2×v;(4)×,产=xi+y+zk,是位置矢量 解:八.=小=小小+小5+小3小 ++2 30 v·W=vWw1=v11+v22+v33=1×3-2×1-5×1=-4 ×ⅳ三 2-5 =1(2×1-1×5)+j(-3×5-1×1)+k(-1×1-2×3) 3i-16 )J-7k
例题2. 已知, , 求: 是位置矢量. (1).v w; (2).v w; (3). v v; 1 2 (4). ; (4). ; (4). , , e v e v r v r xi yj zk = + + v i j k w i j k = + − = − + 2 5 , 3 . v w = vi wi = v1 w1 + v2 w2 + v3 w3 =13− 21− 51 = −4 1 2 5 3 1 1 (2 1 1 5) ( 3 5 1 1) ( 1 1 2 3) 3 16 7 i j k v w i j k i j k = − − = − + − − + − − = − − − 1 2 5 30 2 2 2 v v = vi vi = v1 v1 + v2 v2 + v3 v3 = + + = 解 :
F×=x (-5y-2=)+(+5x)j+(2x-y)k 2-5 11=i(i+2j-5k)= 1v=l1·ve1=,v;=v1 e1×v=e1xve1= Livie 6i3y2e3+6122=v,k-v2j=2k+5 e1×节=i×(+2j-5k)=27×j-5×k=2k+5
e1 v = e1 v j e j = 1 j v j = v1 = 1 e 1 v = i (i + 2 j − 5k ) = 1 v e v e v k v j k j e v e v e v e y z i i i j i j 123 2 3 132 3 2 2 5 1 1 1 = + = − = + = = e v i i j k i j i k k j 1 = ( + 2 − 5 ) = 2 − 5 = 2 + 5 x y z y z i z x j x y k i j k r v ( 5 2 ) ( 5 ) (2 ) 1 2 5 = − − + + + − − =
例题3.证明×(×)=v(·)-v(·v 证明:a×(×)=E1Mu,(v× E.u.& v,w ik imk mnO)n1厘nyv v.:v:-v::v =v(·w)-(l·v)
例题3. 证明 证明: u (v w) v(u w) w(u v) = − ijk j klm l m ijk lmk j l m ijk j k u v w u v w u v w u v w = = ( ) = ( ) ( ) ( ) ( ) v u w w u v v u w wu v u v w i j j i j j il jm im jl j l m = − = − = −