Vectors question: Prove that XYZ is a straight line

[shuphie]

MY WORKING:

Why I'm confused: (red lines)
I worked out AB as -OA + OB = -a + 3b = 3b-a
However, the mark scheme uses a-3b and that's how it then gets XY = b + 1/2 (AB) = b - 1/2 (a-3b) = b - 3b/2 + a/2 seen as it should be BY not YB, right?
Using 3b-a would have got me the wrong solutions (they would not all factorise to a-b)
Why do they use -3b+a?

 

[Dr.Peterson]

Please show us that mark scheme, so we can see, for example, if you are misreading it, or if they have made a typo but meant the right thing. I can't say why someone else did something without seeing what they did.

 

[Agent Smith]

If you can prove ...

$|| \vec {XY} || + || \vec {YZ} || = || \vec {XZ}||$

 

[blamocur]

$\gdef\ve#1{\mathbf #1}$
Posting a solution after 1 week has lapsed.

Using notations $\ve A, \ve B\, \ve X, \ve Y, \ve Z$ for vectors OA, OB, OX, OY and OZ respectively.
$$\begin{gathered} \ve A &=& \ve a\\ \ve B &=& \ve 3\ve b\\ \ve X &=& 2\ve b\\ \ve Y &=& \frac{1}{2}(\ve A+\ve B) = \frac{1}{2}(\ve a + 3\ve b)\\ \ve Z &=& 2\ve a \end{gathered}$$$$\begin{aligned} \ve Y - \ve X &=& \frac{1}{2}\ve a -\frac{1}{2}\ve b\\ \ve Z - \ve X &=& 2\ve a - 2\ve b \end{aligned}$$Thus we have $\ve Z - \ve X\ = 4(\ve Y-\ve X)$, which means that $\ve Y-\ve X$ has the same direction as $\ve Z-\ve X$.