Question

If the vectors $b , c , d$ are not coplanar, then the vector
$( a \times b ) \times( c \times d )+( a \times c ) \times( d \times b )$
$+( a \times d ) \times( b \times c )$ is

• A. parallel to $a$
• B. parallel to $b$
• C. parallel to $c$
• D. perpendicular to $a$

Answer 1

Answer: (A)

The vectors
$( a \times b ) \times( c \times d )=[ a \,c \,d ] b -[ b \,c \,d ] a$
$( a \times c ) \times( d \times b )=[ a \,d\, b ] c -[ c \,d \,b ] a$
and $( a \times d ) \times( b \times c )=[ a\, d \,c ] b -[ a \,d \,b ] c$
$\therefore ( a \times b ) \times( c \times d )+( a \times c ) \times( d \times b )$
$+( a \times d ) \times( b \times c )$
$=[ a\,c \,d ] b -[ b \,c \,d ] a +[ a \,d\, b ] c -[ c \,d \,b ] a$
$+[ a \,d\, c ] b -[ a\, d \,b ] c \quad[\because[ a \,c \,d ]=-[ a \,d\, c ]]$
$=-2[ b \,c\, d ] a$
$\therefore $ Given vector is parallel to $a$.