Question

If $d=a\times (b\times c)+b\times (c\times a)+c\times (a\times b)$, then

• A. $d$ is a unit vector
• B. $d=a+b+c$
• C. $d=0$
• D. $a,b,c$ and $d$ are coplanar

Answer 1

The correct option is C $d=0$

$a\times (b\times c)=(a\cdot c)b-(a\cdot b)c$
$b\times (c\times a)=(b\cdot a)c-(c\cdot b)a$
$c\times (a\times b)=(c\cdot b)ba-(c\cdot a)b$
Since the dot product is commutative ,we have
$d=a\times (b\times c)+b\times (c\times a)+c\times (a\times b)=0$