Question

Any vector $\overrightarrow{r}$ in the plane of two non-zero, non collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ can be expressed uniquely as a linear combination $x\overrightarrow{a}+y\overrightarrow{b}$ of $\overrightarrow{a}and\overrightarrow{b},$ where $x\overrightarrow{a}andy\overrightarrow{b}$ are the components of vector $\overrightarrow{r}$

• A. True
• B. False

Answer 1

The correct option is A True

Isn't the statement conveying the same what fundamental theorem in 2D conveys?
Anyways let's prove the statement.
Let $\overrightarrow{a}and\overrightarrow{b}$ be any two non-zero, non collinear vectors and $\overrightarrow{r}$ be any vector coplanar with$\overrightarrow{a}and\overrightarrow{b}$. Take any point O in the plane of $\overrightarrow{a}and\overrightarrow{b}$.

Let $\overrightarrow{OA}=\overrightarrow{a},\overrightarrow{OB}=\overrightarrow{b}and\overrightarrow{OP}=\overrightarrow{r}$
Clearly OA, OB and OP are coplanar. Through P, draw two lines PM and PN perpendicular to OA and OB respectively meeting OA and OB at M and N respectively. Refer to the figure shown
We have $\overrightarrow{OP}=\overrightarrow{OM}+\overrightarrow{MP}$ (Triangular law for vector addition)
= $\overrightarrow{OM}+\overrightarrow{ON}$ [Since MP = ON and MP$|$$|$ON]$\dots$ (1)
Now $\overrightarrow{OM}$ and $\overrightarrow{OA}$ are collinear vectors,
$\overrightarrow{OM}=x\overrightarrow{OA}=x\overrightarrow{a},$ where x is a scalar
$\overrightarrow{ON}=y\overrightarrow{OB}=y\overrightarrow{b},$ where y is a scalar
Hence from (1)
$\overrightarrow{OP}=x\overrightarrow{a}+y\overrightarrow{b}$ or $\overrightarrow{r}=x\overrightarrow{a}+y\overrightarrow{b}$
Now this is only one part of the statement. We have to prove that this is the unique vector $\overrightarrow{r}$ i.e., uniqueness has to be established as well
If possible let
$\overrightarrow{r}=x\overrightarrow{a}+y\overrightarrow{b}$ and $\overrightarrow{r}=x^{\prime }a+y^{\prime }b$ be two different ways of representing $\overrightarrow{r}$
Then we have $\overrightarrow{xa}+\overrightarrow{yb}=\overrightarrow{x^{\prime }a}+\overrightarrow{y^{\prime }{b}^{\prime }}$
or (x - x') $\overrightarrow{a}$+(y-y')$\overrightarrow{b}=\overrightarrow{O}$
But $\overrightarrow{a}$ and $\overrightarrow{b}$ are not collinear vectors
So x - x' = 0 $\Rightarrow$x = x $\dots$ (2)
and y - y' = 0 $\Rightarrow$ y = y $\dots$ (3)
(For two non collinear vectors $\overrightarrow{a}and\overrightarrow{b}z\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{o}$ is possible only when z = y = 0)
So uniqueness is established. This means that there is only one unique way of representing a vector $\overrightarrow{r}$ in terms of 2 other coplanar vectors $\overrightarrow{a}and\overrightarrow{b}$.