HELP!!!!I have no idea where to start or what to do

[SaxyTimmy]

Hello, I am having trouble proving something. The question is as follows:

Any two independent vectors define a plane and any vector in that plane is a linear combination of those vectors. With this in mind, show that the point P in the plane defined by the non-collinear points A,B,C is given by the vector equation:
OP=aOA+bOB+cOC
where O is the origin and a+b+c=1.

Any help with this question would be amazing. Thanks in advance

SaxyTimmy

 

[SaxyTimmy]

Anyone who saw this problem and was interested in a solution, I finally discovered the solution. It is far more easy than it looks.

Show that the point P in the plane defined by the non-collinear points A,B,C is given by the vector equation:
OP=aOA+bOB+cOC
where O is the origin and a+b+c=1.

Proof:
First we know that the vectors OA OB and OC can all be rewritten as combinations of other vectors.

Therefore Let
OA=OB+BA
and
OC=OB+BC

Sub these in

OP=a(OB+BA)+bOB+c(OB+BC)
OP=aOB+aBA+bOB+cOB+cBC
OP=(a+b+c)OB+aBA+cBC
this resembles the normal equation of a plane when a+b+c=1 because then 1*OB is a point on the plane and BA and BC are independent vectors found on the plane.


Therefore, the point P in the plane defined by the non-collinear points A,B,C is given by the vector equation: OP=aOA+bOB+cOC where O is the origin and a+b+c=1.