Displacement Vector Calculator

Displacement Vector Calculator - Calculate Distance & Direction

📐 Calculate displacement vector, magnitude, direction angle, and components. Visualize motion in 2D and 3D space with step-by-step explanations.

What is Displacement?

Displacement is a vector quantity that represents the change in position of an object. It's the straight-line distance from the initial position to the final position, regardless of the actual path taken.

Displacement Formula

2D Displacement:

Δr = ⟨Δx, Δy⟩ = ⟨x₁ - x₀, y₁ - y₀⟩

3D Displacement:

Δr = ⟨Δx, Δy, Δz⟩ = ⟨x₁ - x₀, y₁ - y₀, z₁ - z₀⟩

Magnitude (Distance)

2D:

|Δr| = √(Δx² + Δy²)

3D:

|Δr| = √(Δx² + Δy² + Δz²)

Direction Angle

2D Angle from X-axis:

θ = arctan(Δy/Δx)

• Measured counterclockwise from positive X-axis
• Range: -180° to +180° (or 0° to 360°)
• Use atan2(Δy, Δx) for correct quadrant

Unit Vector

Unit vector has magnitude of 1 and points in the direction of displacement:

û = Δr / |Δr|

Calculation Examples

Example 1: Horizontal Motion

• Initial: (0, 0), Final: (10, 0)
• Displacement: ⟨10, 0⟩ m
• Magnitude: 10 m
• Direction: 0° (East)

Example 2: Diagonal Motion

• Initial: (2, 1), Final: (7, 5)
• Displacement: ⟨5, 4⟩ m
• Magnitude: √(5² + 4²) = 6.40 m
• Direction: arctan(4/5) = 38.66°

Example 3: 3D Motion

• Initial: (1, 2, 3), Final: (4, 6, 8)
• Displacement: ⟨3, 4, 5⟩ m
• Magnitude: √(3² + 4² + 5²) = 7.07 m

Displacement vs Distance

Key Differences:

• Displacement: Vector (magnitude + direction), straight-line, can be zero
• Distance: Scalar (magnitude only), total path, always positive

Example: If you walk 5m East, then 5m West:

• Distance traveled: 10 m
• Displacement: 0 m (back at start)

Real-World Applications

• Navigation: GPS calculates displacement to destination
• Physics: Velocity = displacement / time
• Engineering: Structural deformation analysis
• Robotics: Path planning and position tracking
• Sports: Analyzing player movement on field
• Aviation: Flight paths and navigation

Compass Directions

• N (North): 90° (positive Y-axis)
• E (East): 0° (positive X-axis)
• S (South): -90° or 270° (negative Y-axis)
• W (West): ±180° (negative X-axis)
• NE: 45°, SE: -45°, SW: -135°, NW: 135°

Vector Addition

Multiple displacements can be added:

Δr_total = Δr₁ + Δr₂ + Δr₃ + ...

Example: Walk 3m East, then 4m North

• Δr₁ = ⟨3, 0⟩, Δr₂ = ⟨0, 4⟩
• Total: ⟨3, 4⟩, Magnitude: 5m, Direction: 53.13° (NE)

Important Properties

• Displacement magnitude ≤ Distance traveled (equality only for straight path)
• Displacement can be negative (components can be negative)
• Zero displacement ≠ no motion (you can return to start)
• Displacement is path-independent (only start and end matter)

Common Mistakes

• Confusing displacement with distance - they're different!
• Wrong angle quadrant - use atan2 function for correct angle
• Forgetting direction - displacement is a vector, must have direction
• Adding scalars to vectors - can't add distance to displacement

💡 Pro Tip: When solving physics problems, always draw a diagram showing initial and final positions. Use the right-hand rule for 3D problems: thumb (X), index finger (Y), middle finger (Z). Remember that displacement depends ONLY on start and end points, not the path taken - this is why a person walking in a circle has zero displacement despite traveling a large distance!