Question

If $\overline{a}$, $\overline{b}$, $\overline{c}$ are three non-coplanar vectors and $\overline{p}$, $\overline{q}$, $\overline{r}$ are reciprocal vectors, then $(l\overline{a}+m\overline{b}+n\overline{c})(l\overline{p}+m\overline{q}+n\overline{r})=$

• A. $l+m+n$
• B. $l^3+m^3+n^3$
• C. $l^2+m^2+n^2$
• D. None of these

Answer 1

The correct option is C $l^2+m^2+n^2$

Given

$\overrightarrow{a}=\frac{1}{\overrightarrow{p}}$

$\overrightarrow{a}\cdot \overrightarrow{p}=1$

$\overrightarrow{b}=\frac{1}{\overrightarrow{q}}$

$\overrightarrow{b}\cdot \overrightarrow{q}=1$

$\overrightarrow{c}=\frac{1}{\overrightarrow{r}}$

$\overrightarrow{c}\cdot \overrightarrow{r}=1$

$(l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c})\cdot (l\overrightarrow{p}+m\overrightarrow{q}+n\overrightarrow{r})$

$l^2\overrightarrow{a}\cdot \overrightarrow{p}+m^2\overrightarrow{b}\cdot \overrightarrow{q}+n^2\overrightarrow{c}\cdot \overrightarrow{r}$

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