Question

Let $a,b$ and $c$ be a system of three non-coplanar vectors. Then the system $a^{\prime },b^{\prime }$ and $c^{\prime }$ which satisfies $a\cdot a^{\prime }=b\cdot b^{\prime }=c\cdot c^{\prime }=1$ and $a\cdot b^{\prime }.=a\cdot c^{\prime }=b\cdot a^{\prime }.=b\cdot c^{\prime }=c\cdot a^{\prime }=c\cdot b^{\prime }=0$ is called the reciprocal system to the vectors $a,b$ and $c$.

If $a^{\prime },b^{\prime },c^{\prime }$ is a reciprocal system of $a,b,c$ then

• A. $\left[abc\right]\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]=|a\times a^{\prime }||b\times b^{\prime }||c\times c^{\prime }|$
• B. $\left[abc\right]\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]=1$
• C. $a\times a^{\prime }=b\times b^{\prime }=c\times c^{\prime }$
• D. $\left[abc\right]\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]<1$

Answer 1

Answer: (B)

$\left[abc\right]\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]=1$

Since by definition of reciprocal systems $b\cdot a^{\prime }=c\cdot a^{\prime }=0,a^{\prime }$ is perpendicular to both $b$ and $c,$ and hence parallel to $b\times c.$ so $a^{\prime }=t(b\times c)$ Also,$a\cdot a^{\prime }=1,$
So $1=a\cdot a^{\prime }=ta\cdot (b\times c)$
$\Rightarrow t=\frac{1}{a\cdot (b\times c)}=\frac{1}{\left[abc\right]}$
Thus $a^{\prime }=\frac{b\times c}{\left[abc\right]}$ Similarly
$b^{\prime }=\frac{c\times a}{\left[abc\right]}$ and
$c^{\prime }=\frac{a\times b}{\left[abc\right]}$
$\Rightarrow a\times a^{\prime }+b\times b^{\prime }+c\times c^{\prime }$
$\frac{a\times (b\times c)+b\times (c\times a+c\times (a\times b)}{\left[abc\right]}$
Now $a\times (b\times c)=(a\cdot c)b-(a\cdot b)c,b\times (c\times a)=(b\cdot a)c-(b\cdot c)a$
and $c\times (a\times b)=(b\cdot c)a-(a\cdot c)b$
$\therefore a\times (b\times c)+b\times (c\times a)+c\times (a\times b)=0$
So $a\times a^{\prime }+b\times b^{\prime }+c\times c^{\prime }=0$
Next $\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]=(a^{\prime }\times b^{\prime })c^{\prime }$
$=\frac{1}{[abc{]}^3}\left[\right(b\times c)\times (c\times a\left)\right]\cdot (a\times b)$
$=\frac{1}{[abc{]}^3}\left[\right(e\cdot a)c-(e\cdot c\left)a\right]\cdot (a\times b)[e=b\times c]$
$=\frac{1}{[abc{]}^3}\left[\left\{\right(b\times c)\cdot a\}c\right]\cdot (a\times b)$
$=\frac{1}{[abc{]}^3}\left[\right(b\times c)\cdot a][c\cdot (a\times b\left)\right]$
$=\frac{1}{[abc{]}^3}[a\cdot (b\times c\left)\right]\left[\right(a\times b)\cdot c]$
$=\frac{1}{\left[abc\right]}$
We can also show that
$a=\frac{b^{\prime }\times c^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]},b=\frac{c^{\prime }\times a^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]},c=\frac{a^{\prime }\times b^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]},$