IIT JEE , AIEEE Forum - Mathematics , Vectors - Radon challenger series Part-6

Radon222

Consider three non coplanar vectors $\vec{b}, \vec{c}$ and $\vec{b}\times \vec{c}$ .Any vector can be written as:

$\vec{a}=x \vec{b}+y \vec{c}+z (\vec{b} \times \vec{c})$ ........(i)

Taking dot product with $\vec{b} \times \vec{c}$ in eq. (i)

$\vec{a}.(\vec{b} \times \vec{c})=z|\vec{b} \times \vec{c}|^2 \\ \\ \Rightarrow z=\frac{\vec{a}.(\vec{b} \times \vec{c})}{|\vec{b} \times \vec{c}|^2}$

Taking dot product with $\vec{b}$ in eq. (i)

x = $\vec{a}.\vec{b}$

Taking dot product with $\vec{c}$ in eq. (i)

y = $\vec{a}.\vec{c}$

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

Thus the blank is to be filled by $\vec{a}$

Ask & Discuss Questions with Community & Experts