Question

If $\overline{a}$ and $\overline{b}$ are any two non-zero and non-collinear vectors, then prove that any vector $\overline{r}$ coplanar with $\overline{a}$ and $\overline{b}$ can be uniquely expressed as $\overline{r}=t_{1}a+t_{2}b$, where $t_1$ and $t_2$ are scalars.

Answer 1

Let $\overrightarrow{OA}=\overrightarrow{a}$ and $\overrightarrow{OB}=\overrightarrow{b}$ be two non zero and non-collinear vectors.
Let $\overrightarrow{OP}=\overrightarrow{r}$ be any vector coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$. Through $P$, draw $PM$ and $PN$ parallel to $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively. Then, from $△OMP$,
$\overrightarrow{PO}=\overrightarrow{OM}+\overrightarrow{MP}$ [by triangle law of vector addition]
$\Rightarrow \overrightarrow{r}=t_1\overrightarrow{a}+t_2\overrightarrow{b}$, where $t_1$ and $t_2$ are suitable scalars.
If $\overrightarrow{a}\parallel \overrightarrow{b}$, then $\overrightarrow{a}=m\overrightarrow{b}$ [where $m$ is a suitable scalar].