Vector extended response ratio

[Martoman]

Again, I will probably answer this question in under .002 nanoseconds after posting, but still: this is annoying a few of the year 12's and thus is passed on to me... which I can't do... grr

So here it is:

The points A and B have position vectors a and b respectively, referrred to an origin O. The point C lies on AB, between AB, and is such that AC:CB = 2: 1, and D is the midpoint of OC. The line AD produced meets OB at E.

Find the ratios OE:EB and AE:ED. I think that all that I need to know is how to get OE:EB then logic my way through the next one.

I reason that OB = b then OE = kb, with and by going all the way around the triangle formed EB = -kb + a + -a + b = b(1-k) and now the ratio is

.... I am missing something here. Going around the triangle to find OB and EB in different forms has only led me back to them being the same thing.

[qshyrn]

let DE=k AE
let OE=s OB=s b

AC=2/3 AB= 2/3 b- 2/3 a

OD=1/2 (a+ 2/3 b- 2/3 a)

AE=s b - a
DE= k AE=ks b - k a

OD+DE=OE
1/2(a+2/3 b- 2/3 a)+ks b- k a= s b
a(1/2- 2/6-k)+ b(2/6+ks-s)=0
k= 1/2-2/6= 3/6
and
2/6+(3/6)s-s=0
then solve for that and you get k=1/6, s=2/5
Note: its like likely i screwed up somewhere(and yes my solution is quite messy lol)

[Chavi]

Ok - here's my working:

let


and







Okay, so let
 
and


but
and

equate the coefficients:
and  
therefore

therefore

from earlier  
so

and the ratio of

Hope that helps /// Apologies in advance for any errors.

[kamil9876]

I had the same idea in my working before qshyrn beat me to it.

It is true under the assumption that and are not parralel. Which is probably reasonable to assume, otherewise those ratios will not exist.

[Martoman]

Okay, so let
 
and


Hope that helps /// Apologies in advance for any errors.

That last line should read

Then

Which means

so gets it into the right parts as

Many thanks for the assistance, key insight being make another vector to be equal to it. 

[Martoman]

therefore

Also on this line, you should write it as:

so , not as that has 6 parts. I usually think of it as a jumping bean along the line, and then you move 4 equal jumps and then get one more.