OPQR is a rectangle and O is the origin, M is the midpoint of RQ and PT: TQ = 2:1. Find, in terms of p and/or r, in its simplest form (i) MO, (ii) MT, (iii) OT, (iv) MT = (2k/-k) and |MT| = sqrt(180). Find the positive value of k.

Since O P Q R is a rectangle, $\vec{OR} = \mathbf{r}$ and $\vec{RQ} = \vec{OP} = \mathbf{p}$.
M is the midpoint of RQ, so $\vec{RM} = \frac{1}{2}\vec{RQ} = \frac{1}{2}\mathbf{p}$.
We can write $\vec{OM} = \vec{OR} + \vec{RM}$.

$\vec{MO}$ is the negative of $\vec{OM}$.

We can write $\vec{MT} = \vec{MQ} + \vec{QT}$.
M is the midpoint of RQ, so $\vec{MQ} = \frac{1}{2}\vec{RQ} = \frac{1}{2}\mathbf{p}$.
Given PT : TQ = 2 : 1, this means $\vec{QT} = \frac{1}{3}\vec{QP}$.
Since O P Q R is a rectangle, $\vec{QP} = -\vec{PQ} = -\vec{OR} = -\mathbf{r}$.
So, $\vec{QT} = \frac{1}{3}(-\mathbf{r}) = -\frac{1}{3}\mathbf{r}$.

We can write $\vec{OT} = \vec{OP} + \vec{PT}$.
Given $\vec{OP} = \mathbf{p}$.
Given PT : TQ = 2 : 1, this means $\vec{PT} = \frac{2}{3}\vec{PQ}$.
Since O P Q R is a rectangle, $\vec{PQ} = \vec{OR} = \mathbf{r}$.
So, $\vec{PT} = \frac{2}{3}\mathbf{r}$.

The magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$.
For $\vec{MT} = \begin{pmatrix} 2k \\ -k \end{pmatrix}$, its magnitude is:

We are given $|\vec{MT}| = \sqrt{180}$.

Square both sides of the equation:

Take the square root of both sides:

The question asks for the positive value of $k$.