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# P3 3D Vectors watch

1. Just spent hours trying to sove this one, can anybody assist please?
The points A and B have position vectors i + 2j + 2K and 4i + 3j respectively.

A point C divides AB in the ratio λ : 1-λ
Find an expression for OC.

Book answer is: {1+3λ}i + {2 + λ}j + 2{1-λ}k
2. Just spent hours trying to sove this one, can anybody assist please?
The points A and B have position vectors i + 2j + 2K and 4i + 3j respectively.

A point C divides AB in the ratio λ : 1-λ
Find an expression for OC.
OA = i + 2j + 2k

OB = 4i + 3j

AB = AO + OB = -i -2j -2k +4i +3j = 3i + j - 2k

AC = λ(3i + j -2k) = 3λi +λj -2λk

OA + AC = OC = {3λ + 1}i + {λ + 2}j + {2 - 2λ}k
3. (Original post by xlaser31)
Just spent hours trying to sove this one, can anybody assist please?
The points A and B have position vectors i + 2j + 2K and 4i + 3j respectively.

A point C divides AB in the ratio λ : 1-λ
Find an expression for OC.

Book answer is: {1+3λ}i + {2 + λ}j + 2{1-λ}k

C is clearly between AB
The question is telling us that AC: CB are in the ratio λ:1-λ
That is, AC(1-λ)=CB(λ)
(1-λ)[OC-OA]=λ(OB-OC)
(1-λ)(OC)-(1-λ)(OA)=λ(OB)-λ(OC)
OC(1-λ+λ)=λ(OB)+(1-λ)(OA)
OC=λ(4i+3j+0k)+(1-λ)(i+2j+2k)
OC=i(4λ+1-λ)+j(3λ+2-2λ)+k(2-2λ)
OC=i(1+3λ)+j(2+λ)+k[2(1-λ)]

If you click 'reply', as if you were to quote me, then the above is readable.

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