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
$(a\times b)\times (c\times d)=[acd]b-[bcd]a$
$(a\times c)\times (d\times b)=[adb]c-[cdb]a$
and $(a\times d)\times (b\times c)=[adc]b-[adb]c$
$\therefore (a\times b)\times (c\times d)+(a\times c)\times (d\times b)$
$+(a\times d)\times (b\times c)$
$=[acd]b-[bcd]a+[adb]c-[cdb]a$
$+[adc]b-[adb]c[\because [acd]=-[adc]]$
$=-2[bcd]a$
$\therefore$ Given vector is parallel to $a$.