Hello josephk Originally Posted by
josephk A question on vector geometry
For easier reference, I've attached the image file in your Word document as a .jpg.
Here are some suggestions. I'll leave some of the details to you.
(a) (i) $\displaystyle \vec{PQ} = \vec{PO}+\vec{OQ}$
$\displaystyle =-\vec{OP}+\vec{OQ}$
$\displaystyle = ...$ ?
(ii)$\displaystyle \vec TU = \vec{OP}$
(b)$\displaystyle \vec{QM} = \tfrac12\vec{QR}$
$\displaystyle = \tfrac12...$ ?
$\displaystyle \vec{TM} = \vec{TQ} + \vec{QM}$$\displaystyle = 2 ... + \tfrac12 ...$ ?
$\displaystyle = ...$ ?
(c)$\displaystyle PQ:QZ = 3:2$
$\displaystyle \Rightarrow \vec{QZ} = \frac23\vec{PQ}$ $\displaystyle = ...$ ? (Use your answer to (a) (i))
$\displaystyle \vec{TZ} = \vec{TQ} + \vec{QZ}$$\displaystyle = ...$ ? (Express this in terms of $\displaystyle \vec p$ and $\displaystyle \vec q$.)
You should now find that $\displaystyle \vec{TZ}$ is a scalar multiple of $\displaystyle \vec{TM}$. This proves that $\displaystyle T,\; M$ and $\displaystyle Z$ are collinear.
Can you fill in the gaps?
Grandad