How to Find λ in a Triangle with Given Vectors OA, OB, P, and Q?

[songoku]

Homework Statement

Given that:
OA = 5s
OB = 10 t
P is on the middle of OA and AQ = 3 QB

Find λ if BG = λ BP

Homework Equations

basic vector

The Attempt at a Solution

I have found AB, BQ, OQ, BP; all in terms of s and t. How to find λ? I've tried but always came back to equation BP = BP or 0 = 0. Please give me hint to start

Thanks

 

[HallsofIvy]

If you have found AB then you must have some information you have not given here.
The length AB depends not only on the lengths of OA and OB but also on the angle between them.

(There is no "B" in your picture. I assume it is the third vertex of the triangle.

 

[songoku]

If you have found AB then you must have some information you have not given here.
The length AB depends not only on the lengths of OA and OB but also on the angle between them.

(There is no "B" in your picture. I assume it is the third vertex of the triangle.

Yes, the third vertex is B. This is my work:
AB = AO + OB = -5 s + 10 t

Sorry I didn't explain it clearly. I didn't find the length but I found the vector.

BQ = 1/4 BA = 1/4 (5s - 10t)

OQ = OA + AQ = 5 s + 3/4 (10t - 5s) = 5/4 (6t + s)

BP = BO + OP = -10t + 5/2 s

By the way, the value of λ should be 2/5. Thanks

 

[madchap]

Yes, the third vertex is B. This is my work:
AB = AO + OB = -5 s + 10 t

Sorry I didn't explain it clearly. I didn't find the length but I found the vector.

BQ = 1/4 BA = 1/4 (5s - 10t)

OQ = OA + AQ = 5 s + 3/4 (10t - 5s) = 5/4 (6t + s)

BP = BO + OP = -10t + 5/2 s

By the way, the value of λ should be 2/5. Thanks

That looks to be a good start. Now, let OG=μ OQ, then OQ = OA + AQ and after some algebra work. OQ = 5/4 s + 30/4 t

BG = λ [5/2 s - 10t ]

OG + GB = OB
5/4 μs + 30/4 μt + 10 λt - 5/2 λs = 10t and by comparing...

30/4 μ + 10 λ = 10 --- 1
5/4 μ -5/2 λ =0 ---2

Now you are left to solve the simultaneous equations. Hope that helps.

 

[songoku]

Ah damn, didn't think about elimination.

Thanks. That helps a lot