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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: /0 on December 07, 2008, 07:43:19 pm

Title: vectors
Post by: /0 on December 07, 2008, 07:43:19 pm: P, Q and R are points with position vectors $2a-b$ , $3a+b$ , and $a+4b$ (cbf with tildes) respectively relative to an origin O where $a$ and $b$ are non-zero, non-parallel vectors. Given that S is the point on OP produced such that $\vec{OS} = k\vec{OP}$ and $\vec{RS} = m\vec{RQ}$ , evaluate k and m.

Does this seem like a flawed question to you? I drew the vectors on a graph and it seems R, Q, and S are not collinear.

Thanks
Title: Re: vectors
Post by: Collin Li on December 07, 2008, 10:35:52 pm: $\vec{OS} = k(2\vec{a} - \vec{b})$

$\vec{RS} = \vec{OS} - \vec{OR} = (2k-1)\vec{a} - (k+4)\vec{b}$

$\vec{RQ} = \vec{OQ} - \vec{OR} = 2\vec{a} - 3\vec{b}$

Since $\vec{RS} = m\vec{RQ}$

$\implies (2k-1)\vec{a} - (k+4)\vec{b} = 2m\vec{a} - 3m\vec{b}$

$\vec{a}$ and $\vec{b}$ are non-zero, non-parallel vectors, which implies they are linearly independent.

(This must be true, for otherwise, $\alpha\vec{a} + \beta\vec{b} = 0$ has a solution for $\alpha \neq 0$ and $\beta \neq 0$ , which would imply $\vec{a} = -\frac{\beta}{\alpha}\vec{b}$ . This means that the two vectors are parallel, so by contradiction, they cannot be linearly dependent)

Since $\vec{a}$ and $\vec{b}$ are linearly independent, equating coefficients yields:

(1): $2k-1 = 2m$

(2): $k+4 = 3m$

Solving them: $\implies m = 3,\, k = 5$
Title: Re: vectors
Post by: /0 on December 08, 2008, 09:42:47 am: thanks coblin! (2) is meant to be $k+4 = 3m$ but yeah now I understand the method
Title: Re: vectors
Post by: Collin Li on December 08, 2008, 11:19:43 am: Quote from: DivideBy0 on December 08, 2008, 09:42:47 am
thanks coblin! (2) is meant to be $k+4 = 3m$ but yeah now I understand the method

Ah, always messing up one step. :) Fixed
Title: Re: vectors
Post by: hard on December 08, 2008, 07:11:53 pm: COLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLINNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN
Title: Re: vectors
Post by: Collin Li on December 08, 2008, 07:25:22 pm: what is up mate
Title: Re: vectors
Post by: hard on December 08, 2008, 07:27:31 pm: just the ceiling
Title: Re: vectors
Post by: Fyrefly on December 08, 2008, 09:00:53 pm: The sky.
Title: Re: vectors
Post by: Flaming_Arrow on December 08, 2008, 09:08:45 pm: sick kentz