Question

If $a , b$ and $c$ are three non-coplanar vectors and $p , q$ and $r$ are vectors defined by $p =\frac{ b \times c }{[ a b c ]}, q =\frac{ c \times a }{[ a b c ]}$ and $r =\frac{ a \times b }{[ a b c ]}$, then the value of $( a + b ) \cdot p +( b + c ) \cdot q +( c + a ) \cdot r$ is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (D)

3

Answer 2

Let $T_{1} =\left( a + b\right) � p$
$=a �p+b�p$
$a\cdot\frac{b\times c}{\left[a\,b\,c\right]}+\frac{b\cdot\left(b\times c\right)}{\left[a\,b\,c\right]}$
$\frac{\left[a\,b\,c\right]}{\left[a\,b\,c\right]}+\frac{\left[b\,b\,c\right]}{\left[a\,b\,c\right]} = 1 + 0 = 1$
Similarly, $T_{2} = \left(b + c\right). q = 1$
and $T_{3} =\left(c+a\right) �r =1$
$\therefore T_{1}+T _{2} +T _{3} =3$