Question

If the volume of the parallelopiped formed by three non-coplanar vectors $a,b$ and $c$ is 4 cu units, then $[a\times bb\times cc\times a]$ is equal to

• A. 64
• B. 16
• C. 4
• D. 8

Answer 1

Answer: (B)

In scalar triple product, the position of dot and cross can be changed provided the cyclic order is maintained i.e.,
$[abc]=(a\times b)\cdot c=a\cdot (b\times c)$
Put $c\times a=n$
$\therefore [a\times bb\times cc\times a]=(a\times b)\cdot \{(b\times c)\times n\}$
$=(a\times b)\cdot \{(n\cdot b)c-(n\cdot c)b\}$
$=(a\times b)\cdot [\{(c\times a)\cdot b\}c-\{(c\times a)\cdot c\}b]$
$=(a\times b)\cdot \{[cab]c-[cac]b\}$
$=[(a\times b)\cdot c][cab]-0$
$=[abc][abc]$
$=[abc{]}^2=(4{)}^2=16$
Alternate Method
By properties of Scalar triple product,
$[a\times bb\times cc\times a]=[abc{]}^2$
$=(4{)}^2=16$