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Q.
If the volume of the parallelopiped formed by three non-coplanar vectors $a , b$ and $c$ is 4 cu units, then $[ a \times b \,b \times c c \times a ]$ is equal to
In scalar triple product, the position of dot and cross can be changed provided the cyclic order is maintained i.e.,
$[ a b c ]=( a \times b ) \cdot c = a \cdot( b \times c )$
Put $c \times a = n$
$\therefore [ a \times b b \times c c \times a ]=( a \times b ) \cdot\{( b \times c ) \times n \} $
$=( a \times b ) \cdot\{( n \cdot b ) c -( n \cdot c ) b \} $
$=( a \times b ) \cdot[\{( c \times a ) \cdot b \} c -\{( c \times a ) \cdot c \} b ] $
$=( a \times b ) \cdot\{[ c \,a \,b ] c -[ c\, a\, c ] b \} $
$=[( a \times b ) \cdot c ][ c \,a\, b ]-0 $
$=[ a \,b\, c ][ a \,b\, c ] $
$=[ a \,b\, c ]^{2}=(4)^{2}=16$
Alternate Method
By properties of Scalar triple product,
$[ a \times b \,b \times c\, c \times a ] =[ a \,b\, c ]^{2} $
$=(4)^{2}=16$