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# Vectors - Bisecting help watch

1. This example is given in the Pure Mathematics book of Year 1/AS Edexcel.

OABC is a parallelogram. P is the point where the diagonals OB and AC intersect. The vectors a and c are equal to OA and OC respectively.

Prove that the diagonals biscect each other.

Working:
OB = OC + CB = c + a
and AC = AO + C
= -OA +OC = -a+c
P lies on OB --> OP = λ(c+a)
P lies on AC --> OP = OA + AP
= a + μ(-a+c)

---> λ(c+a) = a + μ(-a+c)
---> λ = 1-μ and λ=μ
---> λ=μ=1/2, so P is the midpoint of both diagonals, so the diagonals bisect each other.

I understand everything up the point where:
---> λ = 1-μ and λ=μ
takes place, they refer to this as "forming and solving a pair of simulataneous equations by equating the coefficients of a and c" in the book.

Can someone please explain to me how they went from:
---> λ(c+a) = a + μ(-a+c)
to
λ = 1-μ and λ=μ

Thank you!
2. If you try expanding the brackets in both, it will become:

λc+λa = a-μa+μc

Then you can rearrange:

λc+λa = a(1-μ) + μc

Then equating coefficients of a and c:

λ = (1-μ) and λ = μ

Thus:

1-μ = μ

And μ = 1/2 and λ = 1/2
3. (Original post by Truleyhero)
If you try expanding the brackets in both, it will become:

λc+λa = a-μa+μc

Then you can rearrange:

λc+λa = a(1-μ) + μc

Then equating coefficients of a and c:

λ = (1-μ) and λ = μ

Thus:

1-μ = μ

And μ = 1/2 and λ = 1/2
OHHHHHHHHHHH! I understand it now.

Thanks for the help!

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