Given position vectors of M, N, R from the origin are m, n, and r respectively. R is on MN such that MR = 1/4 RN. Find an expression for r in terms of m and n.

The vector from point A to point B is given by $\vec{AB} = \vec{OB} - \vec{OA}$, where $\vec{OA}$ and $\vec{OB}$ are the position vectors of A and B from the origin.

$$\vec{MR} = \vec{OR} - \vec{OM} = \mathbf{r} - \mathbf{m}$$ $$\vec{RN} = \vec{ON} - \vec{OR} = \mathbf{n} - \mathbf{r}$$

The problem states that $\vec{MR} = \frac{1}{4} \vec{RN}$.

$$\mathbf{r} - \mathbf{m} = \frac{1}{4} (\mathbf{n} - \mathbf{r})$$

First, multiply both sides by 4 to eliminate the fraction:

$$4(\mathbf{r} - \mathbf{m}) = \mathbf{n} - \mathbf{r}$$

Distribute the 4 on the left side:

$$4\mathbf{r} - 4\mathbf{m} = \mathbf{n} - \mathbf{r}$$

Gather all terms involving $\mathbf{r}$ on one side and other terms on the other side:

$$4\mathbf{r} + \mathbf{r} = \mathbf{n} + 4\mathbf{m}$$

Combine like terms:

$$5\mathbf{r} = 4\mathbf{m} + \mathbf{n}$$

Finally, divide by 5 to isolate $\mathbf{r}$:

$$\mathbf{r} = \frac{4\mathbf{m} + \mathbf{n}}{5}$$

This can also be written as:

$$\mathbf{r} = \frac{4}{5}\mathbf{m} + \frac{1}{5}\mathbf{n}$$

The expression for $\mathbf{r}$ in terms of $\mathbf{m}$ and $\mathbf{n}$ is: