Question

Show that the four points A, B, C and D with position vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ and $\overrightarrow{d}$ respectively are coplanar if and only if $3\overrightarrow{a}-2\overrightarrow{b}+\overrightarrow{c}-2\overrightarrow{d}=\overrightarrow{0}.$

Answer 1

Necessary Condition: Firstly , let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar vectors. Then, one of them is expressible as a linear combination of the other two. Let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}$ for some scalars $x,y.$ Then,
$\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}$ for some scalars $x,y$.
$\Rightarrow l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}=\overrightarrow{0},$ where $l=x,m=y,n=-1$.
Thus, if $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar vectors, then there exists a scalars $l,m,n$ not all zero simultaneously satisfying $l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}=\overrightarrow{0}$ where $l,m,n$ are not all zero simultaneously.

Sufficient Condition: Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three scalars such that there exists scalars $l,m,n$ not all zero simultaneously satisfying $l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}=\overrightarrow{0}$. We have to prove that $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar vectors.
Now,
$$\begin{gathered} l\overrightarrow{a}+m\overrightarrow{b}+n\overrightarrow{c}=\overrightarrow{0.} \\ \Rightarrow n\overrightarrow{c}=-l\overrightarrow{a}-m\overrightarrow{b}. \\ \Rightarrow \overrightarrow{c}=\left(\frac{-l}{n}\right)\overrightarrow{a}+\left(\frac{-m}{n}\right)\overrightarrow{b}. \end{gathered}$$
$\Rightarrow \overrightarrow{c}$ is a linear combination of $\overrightarrow{a}$ and $\overrightarrow{b}$.
$\Rightarrow \overrightarrow{c}$ lies in a plane $\overrightarrow{a}$ and $\overrightarrow{b}$.
Hence, $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar vectors.