Question

If a, b and c are three non-coplanar vectors, then $(a+b-c)\cdot [(a-b)\times (b-c)$ equals

• A. $0$
• B. $a.b\times c$
• C. $a.c\times b$
• D. $3a.b\times c$

Answer 1

Answer: (B)

$[a+b-c].[(a-b)\times (b-c)$
$=(a+b-c).[a\times b-a\times c-b\times b+b\times c]$
$=a.(a\times b)-a.(a\times c)+a.(b\times c)+b.$
$(a\times b)-b(a\times c)+b.(b\times c)-c.(a\times b)$
$+c.(a\times c)-c.(b\times c)$
$=a.(b\times c)-b.(a\times c)-c.(a\times b)$
$=[abc]-[bac]-[cab]$
$=[abc]+[abc]-[abc]$
$=[abc]=a.(b\times c)$