Question

Express $a,b,c$ in terms of their dot ans vector product.

• A. $a=\frac{a.a}{\left[abc\right]}(b\times c)+\frac{a.b}{\left[abc\right]}(c\times a)+\frac{a.c}{\left[abc\right]}(a\times b)$ ,
  $b=\frac{b.a}{\left[abc\right]}(b\times c)+\frac{b.b}{\left[abc\right]}(c\times a)+\frac{b.c}{\left[abc\right]}(a\times b)$, $c=\frac{c.a}{\left[abc\right]}(b\times c)+\frac{c.b}{\left[abc\right]}(c\times a)+\frac{c.c}{\left[abc\right]}(a\times b)$
  
• B. $a=-\frac{a.a}{\left[abc\right]}(b\times c)+\frac{a.b}{\left[abc\right]}(c\times a)+\frac{a.c}{\left[abc\right]}(a\times b)$ ,
  $b=\frac{b.a}{\left[abc\right]}(b\times c)-\frac{b.b}{\left[abc\right]}(c\times a)+\frac{b.c}{\left[abc\right]}(a\times b)$ ,$c=\frac{c.a}{\left[abc\right]}(b\times c)+\frac{c.b}{\left[abc\right]}(c\times a)-\frac{c.c}{\left[abc\right]}(a\times b)$
• C. $a=\frac{a.a}{\left[abc\right]}(b\times c)-\frac{a.b}{\left[abc\right]}(c\times a)-\frac{a.c}{\left[abc\right]}(a\times b)$ ,
  $b=-\frac{b.a}{\left[abc\right]}(b\times c)+\frac{b.b}{\left[abc\right]}(c\times a)-\frac{b.c}{\left[abc\right]}(a\times b)$, $c=-\frac{c.a}{\left[abc\right]}(b\times c)-\frac{c.b}{\left[abc\right]}(c\times a)+\frac{c.c}{\left[abc\right]}(a\times b)$
• D. none of these

Answer 1

The correct option is A

$a=\frac{a.a}{\left[abc\right]}(b\times c)+\frac{a.b}{\left[abc\right]}(c\times a)+\frac{a.c}{\left[abc\right]}(a\times b)$ ,

$b=\frac{b.a}{\left[abc\right]}(b\times c)+\frac{b.b}{\left[abc\right]}(c\times a)+\frac{b.c}{\left[abc\right]}(a\times b)$, $c=\frac{c.a}{\left[abc\right]}(b\times c)+\frac{c.b}{\left[abc\right]}(c\times a)+\frac{c.c}{\left[abc\right]}(a\times b)$

Let $a$ $=l(b\times c)+m(c\times a)+n(a\times b)$ ...(1)
We have to find the values of $l,m$ and $n.$
Multiply both sides of (1) scalarly by $a.$
$a.a=la.(b\times c)+ma.(c\times a)+na.(a\times b)$

$\Rightarrow a.a=l\left[abc\right]+m\left[aca\right]+n\left[aab\right]=l\left[abc\right].$
$\therefore$ Scalar triple product is zero when two vectors are equal

$\therefore l=\frac{a.a}{\left[abc\right]}$
Similarly, multiplying both sides of (1) salarly by $b$ and $c$, we get $m=\frac{a.b}{\left(abc\right)},n=\frac{a.c}{\left[abc\right]}$
$\therefore a=\frac{a.a}{\left[abc\right]}(b\times c)+\frac{a.b}{\left[abc\right]}(c\times a)+\frac{a.c}{\left[abc\right]}(a\times b)$
Similarly we can write the values $b$ and $c$ as $b=\frac{b.a}{\left[abc\right]}(b\times c)+\frac{b.b}{\left[abc\right]}(c\times a)+\frac{b.c}{\left[abc\right]}(a\times b)$
and $c=\frac{c.a}{\left[abc\right]}(b\times c)+\frac{c.b}{\left[abc\right]}(c\times a)+\frac{c.c}{\left[abc\right]}(a\times b).$