if a = i+2j+3k, b= 2i-j+k, c=i+j-2k how i can verify that across(b cross c)= (a b)c i m getingif - Maths - Vector Algebra

Question

if a = i+2j+3k, b= 2i-j+k, c=i+j-2k how i can verify that
across(b cross c)= (a b)c i m getingif a =across(b cross c)= -9i+3k
and (a b)c = 3i+3j-6k explain me how both r equal
2- how to cal volm of parallelopiped whose continuous edge r
represented by vectors a= 2i-3j+k, b= i-j+2k, c= 2i+j+k explain the
concept of this also

Prasanta · Accepted Answer

(1) The formula should be:-

a→×b→×c→=a→
c→b→ - a→ b→c→

This is called the Vector Triple
Product, or Triple
Product Expansion, or also Lagrange's
Formula

Its verification for the 3 given vectors is as follows:-

LHS=i^+2j^+3k^×2i^-j^+k^×i^+j^-2k=i^+2j^+3k^×i^+5j^+3k=-9i^+3k

RHS=2i^-j^+ki^+2j^+3k^ i^+j^-2k^ - i^+j^-2k^i^+2j^+3k
2i^-j^+k=2i^-j^+k1+2-6 -
i^+j^-2k^2-2+3=-6i^+3j^-3k^-3i^+3j^-6k^=-9i^+3k^

So, LHS=RHS
Hence Proved

(2)

Volume = Area of base x Height
=a→×b→ c→cosϕ=a→×b→
c→

So, the volume of the given parallelopiped =
2i^-3j^+k^×i^-j^+2k^ 2i^+j^+k^=-5i^-3j^+k 2i^+j^+k^
=-10-3+1=12