Question

If $b$ and $c$ are any two non-collinear vectors, and $a$ is any vector, then
$(a\cdot b)b+(a\cdot c)c+\frac{a\cdot (b\times c)}{{|b\times c|}^2}(b\times c)$ is equal to

• A. $a$
• B. $b$
• C. $c$
• D. None of these

Answer 1

The correct option is A $a$

If $\overrightarrow{\alpha },\overrightarrow{\beta },\overrightarrow{\gamma }$ are three non-collinear vectors then, any vector $r$ can be written in terms of $\alpha ,\beta ,\gamma$ as given,

$\overrightarrow{r}=\frac{(r.\alpha )\overrightarrow{\alpha }}{|\alpha {|}^2}+\frac{(r.\beta )\overrightarrow{\beta }}{|\beta {|}^2}+\frac{(r.\gamma )\overrightarrow{\gamma }}{|\gamma {|}^2}$

In this question we have $\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{b\times c}$, Which are non-collinear and a vector $\overrightarrow{a}$, which can be written as ,

$\overrightarrow{a}=\frac{a.b}{|b{|}^2}\overrightarrow{b}+\frac{a.c}{|c{|}^2}\overrightarrow{c}+\frac{a.(b\times c)}{|b\times c{|}^2}\overrightarrow{(b\times c)}$