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 }{[ a b c ]}, q =\frac{ c \times a }{[ a b c ]}, $ and $ r =\frac{ a \times b }{[ a b c ]},$
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)

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