If and are three vectors such that

MATHEMATICS · Subjective · A Level · Mathematics · Vectors · Vectors

Problem

\mathbf a, \mathbf b

and

\mathbf c

are three vectors such that

\left[\mathbf a\ \mathbf b\ \mathbf c\right]=1

, then find the value of

\left[\mathbf a+\mathbf b+\mathbf c\ \mathbf c+\mathbf a\right]+\left[\mathbf a\times\mathbf b\ \mathbf b\times\mathbf c\ \mathbf c\times\mathbf a\right]+\left[\mathbf a+(\mathbf b\times\mathbf c)\ \mathbf b\times(\mathbf c\times\mathbf a)\ \mathbf c\times(\mathbf a\times\mathbf b)\right]