Prove that a (b*c) = (a*b) c , where a,b,c, are any arbitrary vectors and * denotes cross product - Maths - Vector Algebra

Question

Prove that a (b*c) = (a*b) c , where a,b,c, are any arbitrary
vectors and * denotes cross product

Ashwini Kumar · Accepted Answer

Let a⇀=a1i⏜+a2j⏜+a3k⏜b⇀=b1i⏜+b2j⏜+b3k⏜c⇀=c1i⏜+c2j⏜+c3k⏜∴b⇀×c⇀=b1c2k⏜-b1c3j⏜-b2c1k⏜+b2c3i⏜+b3c1j⏜-b3c2i⏜⇒b⇀×c⇀=b2c3-b3c2i⏜+b3c1-b1c3j⏜+b1c2-b2c1k⏜∴a⇀
b⇀×c⇀=a1b2c3-b3c2+a2b3c1-b1c3+a3b1c2-b2c1     
1Now,a⇀×b⇀=a1b2k⏜-a1b3j⏜-a2b1k⏜+a2b3i⏜+a3b1j⏜-a3b2i⏜⇒a⇀×b⇀=a2b3-a3b2i⏜+a3b1-a1b3j⏜+a1b2-a2b1k⏜∴a⇀×b⇀
c⇀=a2b3-a3b2c1+a3b1-a1b3c2+a1b2-a2b1c3⇒a⇀×b⇀
c⇀=a2b3c1-a3b2c1+a3b1c2-a1b3c2+a1b2c3-a2b1c3⇒a⇀×b⇀
c⇀=a1b2c3-a1b3c2+a2b3c1-a2b1c3+a3b1c2-a3b2c1∴a⇀×b⇀
c⇀=a1b2c3-b3c2+a2b3c1-b1c3+a3b1c2-b2c1     
2From 1 and 2, we geta⇀
b⇀×c⇀=a⇀×b⇀
c⇀Hence, proved

Prove that a.(b*c) = (a*b).c , where a,b,c, are any arbitrary vectors and * denotes cross product.

Prove that a.(bc) = (ab).c , where a,b,c, are any arbitrary vectors and * denotes cross product.