Vector Geometry

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The position vectors of P, Q with reference to an origin O are and and M is the point on PQ such that .

a) Prove that the position vector of M is , where

SOLVED!

The vector and the vector where k and l are positive real numbers and and are unit vectors.

b) Prove that the position vector of any point on the internal bisector of has the form .

c) If M is the point where the internal bisector of meets PQ, show that:

.

Please help on questions b) and c)! Thanks

[Mao]

b)
i dont know if this will be adequate enough, but here it is:

if M is a point on the internal bisector (position vector m) in the form , then













QED?

well, i showed it, i didnt exactly "prove" it.... =S

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Thanks Mao, we're not supposed to have done dot products yet tho ;p

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My teacher demonstrated the solution in class:



a)
And from ,









b)

Draw the vectors and . The internal angle bisector does not depend on the magnitude of the rays which bound the angle, but only their directions. Draw another two vectors such that a rhombus is formed with side lengths and . A property that the rhombus has is that its diagonals bisect its angles. The direction of the diagonal of the drawn rhombus is , so the position vector of any point on it must be of the form .

c)

We have two expressions for

1.

2.

Equating, we get





Equating unit vector coefficients:

[Mao]

*in class*

what class is that =S

ur school must be supersaiyan....

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Lol... it is a unusually hard problem from Essential Advanced General Maths, that's why our teacher showed it, and he had to think for a while too :p