📘 Understanding Displacement and Why It’s Always the Shortest Distance

Displacement is one of the most fundamental ideas in physics, but it often causes confusion because it doesn’t care about the path taken — only where you start and where you end.

Let’s break down why displacement is always the shortest distance between two points.

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✅ 1. Displacement = The Straight Line Between Two Points

In physics:

Displacement is defined as the direct straight-line path from the starting point to the ending point.

And in geometry:

A straight line is the shortest distance between any two points.

So displacement must be the shortest possible distance — by definition.

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📐 2. Why No Other Path Can Be Shorter

If you take any route that is not a straight line:

• a curve
• a zig-zag
• a square or circular path
• or any detour

…it will always be longer than the straight line connecting the same two points.

This is a universal geometric fact.

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🧭 3. The Bench Example (with Correct Math Rendering)

Scenario:

The man starts at Point A. The bench is at Point B, 5 meters to the east.

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▶️ If the man walks straight to the bench

• Distance walked = 5 meters
• Displacement = 5 meters east

These two values are the same only because he walked straight.

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▶️ If the man takes a longer path

Say he walks:

• 4 meters north
• 6 meters east
• 4 meters south

He has walked 14 meters total.

But what is his displacement?

He ends at the bench, which is 5 meters east from his starting point.

So:

Displacement = 5 meters east

Even though he walked a lot, his displacement depends only on start and end points, not the path.

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The straight-line distance (i.e., displacement) is:

(5−0)2+(0−0)2=25=5 meters \sqrt{(5-0)^2 + (0-0)^2} = \sqrt{25} = 5 \text{ meters}

This matches the displacement vector:

(5, 0)

No alternate route can be shorter than 5 meters.

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🧠 4. Why the Displacement Vector Automatically Gives the Shortest Path

Displacement is computed as:

d⃗=B−A \vec{d} = B - A

Subtracting coordinates directly constructs the straight line between two points.

This automatically gives:

• the direction
• the shortest possible magnitude
• a path independent of how you actually moved

Displacement ignores all real-world detours.

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🏃 Example: When Distance ≠ Displacement

Imagine someone walks:

• 4 meters north
• 6 meters east
• 4 meters south

Total distance traveled:

14 meters

But their start and end positions might be:

A = (0, 0)
B = (5, 0)

The displacement vector is still:

(5, 0)

And its magnitude is:

∣d⃗∣=5 m |\vec{d}| = 5 \text{ m}

Even though they walked 14 meters, the shortest separation between start and end is still 5 meters.

This is why displacement and distance do not always match.

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🎯 TL;DR

Displacement is the shortest distance because:

1. It is defined as a straight-line path.
2. A straight line is mathematically the shortest route between two points.
3. Coordinate subtraction produces that straight line automatically.
4. Any real-world path you walk will always be equal or longer.

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