已知向量a=(cosa,sina),b=(cosB,sinB),c=(sina,cosa),d=(1⼀2,1⼀3),若a+b=d 求a*b与b*c

是a点乘b吗？以下用(a.b)表示点乘.

解：因为 a=(cos A, sin A),
b=(cos B, sin B),
c=(sin A, cos A),
d=(1/2, 1/3).
所以 a^2=b^2=1,
d^2=13/36.

(i)先求(a.b).
因为 a+b=d,
所以 13/36=d^2
=(a+b)^2
=a^2+2(a.b)+b^2
=2+2(a.b).
解得 (a.b)= -59/72.

(ii)再求(b.c).
由(i)知,
(a.b)=cos A cosB +sin A sin B = -59/72.
即 cos (A-B) = -59/72.
因为 a+b=d,
所以 cos A +cos B =1/2, (1)
sin A +sin B =1/3. (2)
(1)*(2)得
1/6 =(cos A +cos B)(sin A +sin B)
=cos A sin A +cos A sin B
+cos B sin A +cos B sin B
=sin (A+B) +(1/2)(sin 2A +sin 2B)
=sin (A+B) +sin (A+B) cos (A-B)
=(1-59/72)sin (A+B).
解得 sin (A+B) =12/13.
所以 (b.c)=cos B sin A +sin B cos A
=sin (A+B)
=12/13.