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Question

Any vector r in the plane of two non-zero, non collinear vectors a and b can be expressed uniquely as a linear combination xa+yb of a and b, where xa and yb are the components of vector r

A
True
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B
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 a and b be any two non-zero, non collinear vectors and r be any vector coplanar witha andb. Take any point O in the plane of a and b.

Let OA=a,OB=bandOP=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 OP=OM+MP (Triangular law for vector addition)
= OM+ON [Since MP = ON and MP||ON] (1)
Now OM and OA are collinear vectors,
OM=xOA=xa, where x is a scalar
ON=yOB=yb, where y is a scalar
Hence from (1)
OP=xa+yb or r=xa+yb
Now this is only one part of the statement. We have to prove that this is the unique vector r i.e., uniqueness has to be established as well
If possible let
r=xa+yb and r=xa+yb be two different ways of representing r
Then we have xa+yb=xa+yb
or (x - x') a+(y-y')b=O
But a and b are not collinear vectors
So x - x' = 0 x = x (2)
and y - y' = 0 y = y (3)
(For two non collinear vectors aandbza+yb=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 r in terms of 2 other coplanar vectors aandb.

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