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Force and Displacement Vectors Visualization
Problem Statement
Find the angle between force F = (3î + 4ĵ - 5k̂) and displacement d = (5î + 4ĵ + 3k̂). Also find the projection of F on d.
Solution
1. Dot Product Calculation
F·d = Fₓdₓ + Fᵧdᵧ + F₂d₂ = 3(5) + 4(4) + (-5)(3) = 15 + 16 - 15 = 16
2. Magnitude Calculations
|F| = √(3² + 4² + (-5)²) = √(9 + 16 + 25) = √50 ≈ 7.07
|d| = √(5² + 4² + 3²) = √(25 + 16 + 9) = √50 ≈ 7.07
|d| = √(5² + 4² + 3²) = √(25 + 16 + 9) = √50 ≈ 7.07
3. Angle Calculation
cosθ = (F·d)/(|F||d|) = 16/(√50 × √50) = 16/50 = 0.32
θ = cos⁻¹(0.32) ≈ 71.34°
θ = cos⁻¹(0.32) ≈ 71.34°
4. Projection Calculation
Projection of F on d = (F·d)/|d| = 16/√50 ≈ 2.26
Visualization Explanation
The 3D visualization above shows:
- Red arrow: Force vector F = (3, 4, -5)
- Blue arrow: Displacement vector d = (5, 4, 3)
- The angle θ between the vectors is shown in yellow
- The projection of F onto d is shown in green
