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Fundamental Theorem in 2D

Any vector r⃗...

Question

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

True

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False

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Solution

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 $\to a a n d \to b$ be any two non-zero, non collinear vectors and $\to r$ be any vector coplanar with $\to a a n d \to b$ . Take any point O in the plane of $\to a a n d \to b$ .

Let $- - \to O A = \to a, - - \to O B = \to b a n d - - \to O P = \to r$
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 $- - \to O P = - - \to O M + - - \to M P$ (Triangular law for vector addition)
= $- - \to O M + - - \to O N$ [Since MP = ON and MP $|$ $|$ ON] $\dots$ (1)
Now $- - \to O M$ and $- - \to O A$ are collinear vectors,
$- - \to O M = x - - \to O A = x \to a,$ where x is a scalar
$- - \to O N = y - - \to O B = y \to b,$ where y is a scalar
Hence from (1)
$- - \to O P = x \to a + y \to b$ or $\to r = x \to a + y \to b$
Now this is only one part of the statement. We have to prove that this is the unique vector $\to r$ i.e., uniqueness has to be established as well
If possible let
$\to r = x \to a + y \to b$ and $\to r = x^{'} a + y^{'} b$ be two different ways of representing $\to r$
Then we have $- \to x a + \to y b = - \to x^{'} a + - - \to y^{'} b^{'}$
or (x - x') $\to a$ +(y-y') $\to b = \to O$
But $\to a$ and $\to b$ 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 $\to a a n d \to b z \to a + y \to b = \to o$ is possible only when z = y = 0)
So uniqueness is established. This means that there is only one unique way of representing a vector $\to r$ in terms of 2 other coplanar vectors $\to a a n d \to b$ .

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