Question

It $p,q$ and $r$ are non-zero, non-coplanar vectors then $[p+q-rp-qq-r]=\dots$

• A. $3[pqr]$
• B. $0$
• C. $[pqr]$
• D. $2[pqr]$

Answer 1

Answer: (C)

Given, $p,q$ and $r$ are non-zero coplanar vectors.
$\therefore [p+q-rp-qq-r]$
$=(p+q-r)\cdot [(p-q)\times (q-r)]$
$=(p+q-r)\cdot [p\times q-p\times r-q\times q+q\times r]$
$=(p+q-r)\cdot [p\times q-p\times r+q\times r](\because q\times q=0)$
$=p\cdot (p\times q)-p\cdot (p\times r)+p\cdot (q\times r)$
$+q\cdot (p\times q)-q\cdot (p\times r)+q\cdot (q\times r)$
$-r\cdot (p\times q)-r\cdot (p\times r)+r\cdot (q\times r)$
$=[pqr]-[qpr]-[rpq]$
$=[pqr]+[pqr]-[pqr]$
$=2[pqr]-[pqr]=[pqr]$