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Example 12. Let a=i+j-k, bai, > >, , Find the vector a x(b x c)., , ~j+k, c=i-j-k,, , Solution. ax (bx ¢) =(a.c)b-(a. bye, =10+7-%.0-j-O1G-7+b-10+ j-b.G-j+OG-7-], = (1-141) (i-j+h-[-1-1)G-j-), =(i-j+k)+(i-j-h = 27-2] Ans., Example 13. Prove that ;, , ax(bxc)+bx(exa)+ex(ax b)=0 (Nagpur University, Winter 2008), Solution. Here, we have, , ax(bxc)+bx(cexa)+ex(axb), =[(a.c)b-(a. b) J+ (0. a) cc -(B.c) a] +(e .b) B-(C. a) B|, =[(b.a)¢ —(a.b)e]+[(e.b)a -(b.c)a}+[(a.c) b —(c.a) bl, = [(a.b) ce -(a. B) c] +B. c) a - (Bc) a] +(e .a) B -(c a) BY, =0+0+0=0 Proved., Example 14. Prove that :, ix(@ xi)tjx(axptkx(@xk=24 (Nagpur University, Winter 2003), , Solution. Let @ = a,i+a,j+a,k, Now, LHS. = ix(axi)+jx(ax j)+kx(axk), , a, , ", , x (Ce; Pein fa, Bvct | + J x| (a tem, fom, Bx 9], Sula, fon, j+mbxll, ?x[ q(x i) a iy +athkxd]+]x[mdxp+ad=precex})|, Exlqdix B+eG xd onits |, = fx[0-a,£40,)|+}x[qf+0-«, i] +fx|-4 +0240], = —a,(i x R) + a,(i x j) +.,(j x B) - a,(j x i) - a(R x j) + a(h x i), , = mptak+aitaks+a,ita] = 2 i+2a,]+ 2k, , = 2(a,ita,jtak=2a Proved.
