Question

If $\overline{a}$, $\overline{b}$, $\overline{c}$ be three non-coplanar vectors and ${\overline{a}}_1$, ${\overline{b}}_1$, ${\overline{c}}_1$ constitute the corresponding reciprocal system of vectors, then for any arbitrary vector $\overline{r}$.

• A. $\overline{r}=(\overline{r}.{\overline{a}}_1){\overline{a}}_1+(\overline{r}.{\overline{b}}_1){\overline{b}}_1+(\overline{r}.{\overline{c}}_1){\overline{c}}_1$
• B. $\overline{r}=(\overline{r}.\overline{a})\overline{a}+(\overline{r}.\overline{b})\overline{b}+(\overline{r}.\overline{c})\overline{c}$
• C. $\overline{r}=(\overline{r}.{\overline{a}}_1)\overline{a}+(\overline{r}.{\overline{b}}_1)\overline{b}+(\overline{r}.{\overline{c}}_1)\overline{c}$
• D. None of these

Answer 1

The correct option is C $\overline{r}=(\overline{r}.{\overline{a}}_1)\overline{a}+(\overline{r}.{\overline{b}}_1)\overline{b}+(\overline{r}.{\overline{c}}_1)\overline{c}$

Let $\overline{r}=\lambda \overline{a}+\mu \overline{b}+\delta \overline{c}$
$\therefore$ $\overline{r}.{\overline{a}}_1=\lambda$, $\overline{r}.{\overline{b}}_1=\mu$, $\overline{r}.{\overline{c}}_1=\delta$ ..(According to the definition of reciprocal system of vectors)
$\therefore$ $\overline{r}=(\overline{r}.{\overline{a}}_1)\overline{a}+(\overline{r}.{\overline{b}}_1)\overline{b}+(\overline{r}.{\overline{c}}_1)\overline{c}$