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Simplify: Using Vector Algebra: 1/2 (a +b) dot [c cross (2a +b)]

Mathematics
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|dw:1355251229553:dw|
do i distribute the 1/2 or the (a+b)?
if i remember properly, cX(a.b) is the scalar triple product or box product denoted by [a b c] ans a. [a b c] = b. [a b c] = c. [a b c] = 0 so if u distribute (a+b), you'll get final answer as 0

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Other answers:

also [na mb pc] = mnp [a b c ]
*and
so i wouldn't simplify the inside of the brackets?
[cX 2a.b] = 2 [a b c]
i think i mis-interpreted the question....
|dw:1355252100635:dw|
=0 + 1/2 [a c b]+ 1/2 b . 2c X a + 1/2 b . c X b =0 + 1/2 [a c b]+1/2 [b c a]+ 1/2[b c b] =0 -1/2 [a b c] +1/2 [a b c] + 0 =0+0+0=0 now this seems right.
if you could understand.... ask if there is any doubt in any step.
just one question whats the operation inside the bracket i.e [b c a]?
cyclic change doesn't change sign [a b c] = [b c a] = [c a b] else, sign changes [a b c] = -[a c b ] = -[b a c]
if any 2 or more vectors are same, then box product =0 [a a b] = [b b a] = ... = 0
or u meant to ask [a b c] = a.bXc ?
yes
thanks so much!
hmm...welcome ^_^

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