第1章物理学力学数学矢量习题解答 第1章物理学力学数学矢量习题解答 =A×C+B×C+C×A+B×A+A×B+C×B=0 16.已知a=1+2r2)i+ej-,求号, 2②ix(G+R)-jx(+府+Rx+i+R) 解:要=[1+2r2)i+ej-府=4i-e方 =k-j+k-i+j-i=2k-2i =孟(4i-e')=4i+ej (3)(2A+B)×(C-A)+(B+C)×(A+B) 17.已知A=3ei-(4r3-0j+k,B=412i+30, =2Ax(C-A+B×(C-A+B×(A+B)+C×(A+B) =2A×C+B×C-B×A+B×A+C×A+C×B 求(A·B) =A×C 解:AB=AB+A,B,+AB 14.计算下面诸式 =3e4r2-3t(413-t) 解:()i(j×+k(×》+(K×i =12r2e--12r4+3r2 =i.i+k.k+j.j=3 号(4.B)=4(12r2e-12r+32) (2)A(B×A)=B.(A×)=0 =12(21-t2)e-48t3+61 15.求证:(A+B)-[(A+C)×B)]=-A·(B×C) 证明:(A+B)[(A+C)×B】 =A(A×B+C×B)+B-(A×B+C×B) =A(A×B)+A(C×B)+B(A×B)+B.(C×B) =B.(A×A+A(C×B)+A(B×B)+C.(B×) =A(C×B)=-A(B×C)第1章物理学力学数学 矢量习题解答 4 第 1章物理学力学数学 矢量习题解答 = AC + BC + C A + B A + A B + C B = 0 ⑵ ) ˆ ˆ ˆ ( ˆ ) ˆ ˆ ( ˆ ) ˆ ˆ ( i ˆ j + k − j i + k + k i + j + k k j k i j i k i = ˆ − ˆ + ˆ − ˆ + ˆ − ˆ = 2 ˆ − 2 ˆ ⑶ (2A B) (C A) (B C) (A B) + − + + + A C A C B C B A B A C A C B A C A B C A B A B C A B = = + − + + + = − + − + + + + 2 2 ( ) ( ) ( ) ( ) 14.计算下面诸式 解:⑴ ) ˆ ˆ ( ˆ ) ˆ ˆ ( ˆ ) ˆ ˆ ( ˆ i j k + k i j + j k i ˆ ˆ 3 ˆ ˆ ˆ ˆ = i i + k k + j j = ⑵ A(B A) = B(A A) = 0 15.求证: (A B) [(A C) B)] A (B C) + + = − 证明: (A B) [(A C) B)] + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A C B A B C B A A A C B A B B C B B A A B A C B B A B B C B A A B C B B A B C B = = − = + + + = + + + = + + + 16. , . ˆ ˆ ˆ (1 2 ) 2 2 2 dt d A dt t dA A t i e j k 已知 = + + − − ,求 解: t i e j k ti e j t t dt d dt dA 4 ˆ ˆ ] ˆ ˆ ˆ [(1 2 ) 2 − − = + + − = − ti e j i e j t t dt d dt d A 4 ˆ ˆ ) 4 ˆ ˆ 2 ( 2 − − = − = + 17.已知 A e i t t j tk B t i tj t 4 ˆ 3 ˆ , ˆ ˆ (4 ) 3 ˆ 3 2 = − − + = + − , (A B) dt d 求 解: A B = AxBx + AyBy + AzBz 2 4 2 2 3 12 12 3 3 4 3 (4 ) t e t t e t t t t t t = − + = − − − − ( ) (12 12 3 ) 2 4 2 A B t e t t t dt d dt d = − + − t t e t t t 12(2 ) 48 6 2 3 = − − + − j i k j i k