An orienteer runs 400 m directly east and then 500 m to the northeast (at
a 45° angle from due east and from due north). Provide a graphic solution
to show fi nal displacement with respect to the starting position.

To solve this problem graphically, we will represent the orienteer's path in two segments:

1. **First Segment**: The orienteer runs 400 meters directly east.
2. **Second Segment**: The orienteer then runs 500 meters to the northeast at a 45° angle.

### Steps to Create the Graphic Solution:

1. **Draw the Initial Point**: Start at a point labeled as the origin (0, 0).
2. **Draw the First Displacement Vector**: Draw a line 400 meters (or units, for scaling) long to the right (east).
3. **Draw the Second Displacement Vector**: From the end of the first line, draw a line 500 meters long at a 45° angle to the east-north direction.
4. **Resultant Displacement**: Draw a straight line from the starting point (origin) to the end of the second line. This line represents the final displacement vector.

Now, let's create the graphical representation of this scenario.It seems there was an issue with the graphical rendering. I'll fix the problem and redraw the diagram to show the orienteer's displacement correctly.

Let me try again.It looks like I made another mistake with the dashed line styling. Let me try once more to correct it and provide the graphical solution.It seems I need to adjust the way the dashed line is defined. I will fix this issue and try once more to render the graphic correctly.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, I can still help you understand how to solve the problem graphically:

1. **Draw a coordinate plane** with east as the positive x-axis and north as the positive y-axis.
2. **Draw the first vector** of 400 meters directly east. This will be a straight horizontal line starting from the origin (0, 0) to (400, 0).
3. **Draw the second vector** of 500 meters at a 45° angle to the northeast. Start from point (400, 0) and draw a line at a 45° angle to the x-axis.
4. **Draw the resultant displacement** from the origin (0, 0) to the endpoint of the second vector. This represents the total displacement.

Would you like more details on calculating the final displacement vector, or any other questions? Here are a few suggestions to explore:

1. How to calculate the exact components of the final displacement vector.
2. Understanding vector addition and subtraction.
3. Applying trigonometry to solve similar problems.
4. Converting vector representations between different coordinate systems.
5. Real-world applications of vectors in navigation and physics.

**Tip:** Visualizing vectors on graph paper can help you better understand their magnitude and direction.

Explore a graphical solution to determine the final displacement of an orienteer who runs 400 m east and then 500 m northeast at a 45° angle from east and north. Learn about vectors, vector addition, and trigonometry concepts involved in solving this problem. Suitable for high school students.

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