Q46 / 120 · VITEEE 2014 · Mathematics
If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are three non-coplanar vectors and $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are reciprocal vectors, then $(l a+m b+n c)$. $(l \mathrm{p}+m \mathrm{q}+n \mathrm{r})$ is equal to
- A$l+m+n$
- B$l^{3}+m^{3}+n^{3}$
- Correct answerC$l^{2}+m^{2}+n^{2}$
- DNone of these
Solution
$\mathrm{p}, \mathrm{q}$ and $\mathrm{r}$ are reciprocal vectors of $\mathbf{a}, \mathbf{b}$ and c respectively.
$\begin{array}{l}
\text { So, } \boldsymbol{p . r}=1, \boldsymbol{p . b}=0=\boldsymbol{p . c} \\
\boldsymbol{q . a}=0, \boldsymbol{q . b}=1, \boldsymbol{q . c}=0 \\
\boldsymbol{r} \cdot \boldsymbol{a}=0, \boldsymbol{r . b}=0, \boldsymbol{r . c}=1 \\
\therefore(\boldsymbol{l a}+\boldsymbol{m b}+\boldsymbol{n c}) \cdot(\boldsymbol{l p}+\boldsymbol{m q}+\boldsymbol{n} \boldsymbol{r}) \\
=\boldsymbol{l}^{\mathbf{2}}+\boldsymbol{m}^{\mathbf{2}}+\boldsymbol{n}^{\mathbf{2}}
\end{array}$
Hence, the correct answer is option: 'C'.