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 $c$.

Find the correct one:

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

Answer 1

The correct option is B $a=\frac{b^{\prime }\times c^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}$

Given

$\overrightarrow{a}\cdot a^{\prime }=\overrightarrow{b}\cdot b^{\prime }=\overrightarrow{c}\cdot c^{\prime }=1$

$\overrightarrow{a}\cdot b^{\prime }=\overrightarrow{a}\cdot c^{\prime }=\overrightarrow{b}\cdot a^{\prime }=\overrightarrow{b}\cdot c^{\prime }=\overrightarrow{c}\cdot a^{\prime }=\overrightarrow{c}\cdot b^{\prime }=0$

$a+b+c=\frac{1}{a^{\prime }}+\frac{1}{b^{\prime }}+\frac{1}{c^{\prime }}$

by taking LCM

$a+b+c=\frac{b^{\prime }\times c^{\prime }+c^{\prime }\times a^{\prime }+a^{\prime }\times b^{\prime }}{(a^{\prime }\times b^{\prime })\cdot (a^{\prime }\times c^{\prime })}$

$a+b+c=\frac{b^{\prime }\times c^{\prime }+c^{\prime }\times a^{\prime }+a^{\prime }\times b^{\prime }}{a^{\prime }\cdot (b^{\prime }\times c^{\prime })}$

$a+b+c=\frac{b^{\prime }\times c^{\prime }+c^{\prime }\times a^{\prime }+a^{\prime }\times b^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}$

$a+b+c=\frac{b^{\prime }\times c^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}+\frac{c^{\prime }\times a^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}+\frac{a^{\prime }\times b^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}$

comparing both sides

$\overrightarrow{a}=\frac{b^{\prime }\times c^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}$

$b=\frac{c^{\prime }\times a^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}$

$c=\frac{a^{\prime }\times b^{\prime }}{\left[a^{\prime }{b}^{\prime }{c}^{\prime }\right]}$