a.ghetch the diagram. Shoo the pasition of $$ K $$ and $$ S $$ such that: - $$ R $$ is the midpaint of OK - $$ S $$ is a point on $$ O $$ M such $$ \overrightarrow{O S}=1 / 3 \overrightarrow{O m} $$ b. Write down in the vectors i $$ \overrightarrow{m k} $$ ii $$ \overrightarrow{R m} $$ iii $$ \overrightarrow{k S} $$ iv $$ \overrightarrow{\mathrm{hs}} $$ $$ 3 \cdot \overrightarrow{O M} $$ on $$ \overrightarrow{O N} $$ are position vectors with respect to the orgin $$ O $$, such that $$ \overrightarrow{O M}=m $$ and $$ \overrightarrow{O M}=n \quad L $$ is a point on MN such that $$ M L: L N=2: 1 $$ (i) Draw a sketch of the triangle OMIV and label the points $$ O, m, N $$ and $$ L $$.

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UpStudy AI · Accepted Answer

To solve the problem, let's break it down into parts as requested.

### Part a: Diagram and Position of Points

1. **Position of Points**:
   - Let $$ O $$ be the origin.
   - Let $$ M $$ be a point such that $$ \overrightarrow{O M} = m $$.
   - Let $$ K $$ be a point such that $$ R $$ is the midpoint of $$ OK $$. Therefore, we can express $$ R $$ as:
     $$
     \overrightarrow{R} = \frac{1}{2}(\overrightarrow{O} + \overrightarrow{K}) = \frac{1}{2}(0 + \overrightarrow{K}) = \frac{1}{2} \overrightarrow{K}
     $$
   - Let $$ S $$ be a point on $$ OM $$ such that:
     $$
     \overrightarrow{O S} = \frac{1}{3} \overrightarrow{O M} = \frac{1}{3} m
     $$

### Part b: Write Down the Vectors

i. **Vector $$ \overrightarrow{M K} $$**:
   $$
   \overrightarrow{M K} = \overrightarrow{K} - \overrightarrow{M}
   $$

ii. **Vector $$ \overrightarrow{R M} $$**:
   $$
   \overrightarrow{R M} = \overrightarrow{M} - \overrightarrow{R}
   $$

iii. **Vector $$ \overrightarrow{K S} $$**:
   $$
   \overrightarrow{K S} = \overrightarrow{S} - \overrightarrow{K}
   $$

iv. **Vector $$ \overrightarrow{H S} $$**:
   $$
   \overrightarrow{H S} = \overrightarrow{S} - \overrightarrow{H}
   $$

### Additional Information

Given that $$ \overrightarrow{O M} = m $$ and $$ \overrightarrow{O N} = n $$, and $$ L $$ is a point on $$ MN $$ such that $$ ML: LN = 2:1 $$, we can express the position of $$ L $$ as follows:

- The position vector of $$ L $$ can be found using the section formula:
  $$
  \overrightarrow{L} = \frac{2\overrightarrow{N} + 1\overrightarrow{M}}{2 + 1} = \frac{2n + m}{3}
  $$

### Sketch of Triangle $$ OMN $$

To visualize the triangle $$ OMN $$:
- Draw point $$ O $$ at the origin.
- Draw point $$ M $$ at position $$ m $$.
- Draw point $$ N $$ at position $$ n $$.
- Label the points $$ O, M, N, $$ and $$ L $$ accordingly.

### Conclusion

This breakdown provides the necessary vectors and positions based on the given conditions. If you need a diagram or further calculations, please let me know!
To solve the problem, follow these steps: 1. **Position Points**: - Let $$ O $$ be the origin. - Let $$ M $$ be at position $$ m $$. - $$ R $$ is the midpoint of $$ OK $$, so $$ \overrightarrow{R} = \frac{1}{2} \overrightarrow{K} $$. - $$ S $$ is on $$ OM $$ such that $$ \overrightarrow{O S} = \frac{1}{3} m $$. 2. **Vectors**: - $$ \overrightarrow{M K} = \overrightarrow{K} - m $$ - $$ \overrightarrow{R M} = m - \frac{1}{2} \overrightarrow{K} $$ - $$ \overrightarrow{K S} = \frac{1}{3} m - \overrightarrow{K} $$ - $$ \overrightarrow{H S} = \overrightarrow{S} - \overrightarrow{H} $$ 3. **Additional Information**: - $$ L $$ is on $$ MN $$ with $$ ML: LN = 2:1 $$, so $$ \overrightarrow{L} = \frac{2n + m}{3} $$. 4. **Sketch**: - Draw triangle $$ OMN $$ with points $$ O, M, N, $$ and $$ L $$ labeled accordingly. This provides the required vectors and positions based on the given conditions.

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