Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non-coplanar vectors, and $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ are reciprocal vectors to $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ respectively, then $(l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}).(l\overrightarrow{p}+m\overrightarrow{q}+n\overrightarrow{r})$ is equal to$:($where $l,m,n$ are scalars$)$

• A. $l^2+m^2+n^2$
• B. $|m+mn+n|$
• C. $0$
• D. None of these

Answer 1

Answer: (A)

$l^2+m^2+n^2$

we know dot product of :

$(m\overrightarrow{a}+n\overrightarrow{b})(p\overrightarrow{a}+q\overrightarrow{b})=mp\overrightarrow{a}+nq\overrightarrow{b}\Rightarrow (l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}).(l\overrightarrow{p}+m\overrightarrow{q}+n\overrightarrow{r})\Rightarrow l^2.\overrightarrow{a}.\overrightarrow{p}+m^2.\overrightarrow{b}.\overrightarrow{q}+n^2.\overrightarrow{c}.\overrightarrow{r}⟶\left(i\right)\because \overrightarrow{p},\overrightarrow{q}$

and $\overrightarrow{r}$ are reciprocal.

vectors of $\overrightarrow{a,}\overrightarrow{b}$ ,and $\overrightarrow{c}$

respectively $\Rightarrow \overrightarrow{a}.\overrightarrow{p}=1\overrightarrow{b}.\overrightarrow{q}=1\overrightarrow{c}.\overrightarrow{r}=1$

$\Rightarrow \left(i\right)=l^2+m^2+n^2$