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Double vector product

The double vector product is defined as the vector [a,[b,c]], where [a,b] denotes the vector product of vectors a and b.

1. The vector [a,[b,c]] is called the double vector product. Prove that the equality [a,[b,c]]=b(a,c)−c(a,b). holds true.

Solution.

Let a=(ax,ay,az),b=(bx,by,bz),c=(cx,cy,cz). Then

[b,c]=|ijkbxbybzcxcycz|=i(bycz−bzcy)−j(bxcz−bzcx)+k(bxcy−bycx).

[a,[b,c]]=|ijkaxayazbycz−bzcy−bxcz+bzcxbxcy−bycx|= =i(ay(bxcy−bycx)−az(bzcx−bxcz))− −j(ax(bxcy−bycx)−az(bycz−bzcy))+ +k(ax(bzcx−bxcz)−ay(bycz−bzcy))= =i(aybxcy−aybycx+azbxcz−azbzcx)− −j(axbxcy−axbycx−azbycz+azbzcy)+ +k(axbzcx−axbxcz−aybycz+aybzcy).

Next, let's find the vector b(a,c)−c(a,b):

b(a,c)=(bx,by,bz)(axcx+aycy+azcz)= =(axbxcx+aybxcy+azbxcz,axbycx+aybycy+azbycz, axbzcx+aybzcy+azbzcz);

c(a,b)=(cx,cy,cz)(axbx+ayby+azbz)= =(axbxcx+aybycx+azbzcx,axbxcy+aybycy+azbzcy, axbxcz+aybycz+azbzcz);

Hence,

b(a,c)−c(a,b)=i(aybxcy−aybycx+azbxcz−azbzcx)− −j(axbxcy−axbycx−azbycz+azbzcy)+ +k(axbzcx−axbxcz−aybycz+aybzcy)=[a,[b,c]].

Which is what needed to be proved.

Tags: linear algebra, vector, vector product, double vector product

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