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 with
→a and→b. Take any point O in the plane of
→a and →b.
Let
−−→OA=→a,−−→OB=→band−−→OP=→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=x−−→OA=x→a, where x is a scalar
−−→ON=y−−→OB=y→b, where y is a scalar
Hence from (1)
−−→OP=x→a+y→b or
→r=x→a+y→b
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=x→a+y→b and
→r=x′a+y′b be two different ways of representing
→r
Then we have
−→xa+→yb=−→x′a+−−→y′b′
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
→aand→bz→a+y→b=→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
→aand→b.