Question

Let $a,b$ and $c$ be three non-coplanar vectors, and let $p,q$ and $r$ be the vectors defined by the relation
$p=\frac{b\times c}{[abc]},q=\frac{c\times a}{[abc]},$ and $r=\frac{a\times b}{[abc]},$
Then the value of the expression $(a+b)\cdot p+(b+c)\cdot q+(c+a)\cdot r$ is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (D)

$a\cdot p=\frac{a\cdot (b\times c)}{[abc]}=\frac{[abc]}{[abc]}=1$
$b\cdot p=\frac{b\cdot (b\times c)}{[abc]}=\frac{0}{[abc]}=0$
$b\cdot q=\frac{b\cdot (c\times a)}{[abc]}=\frac{(b\times c)\cdot a}{[abc]}\frac{a\cdot (b\times c)}{[abc]}\frac{[abc]}{[abc]}=1$
$c\cdot q=\frac{c\cdot (c\times a)}{[abc]}=0$
$c\cdot r=\frac{(a\times b)\cdot c}{[abc]}=1$ and $a\cdot r=0$
Therefore, the given expression is equal to
$1+0+1+0+1+0=3$.