Vector Addition

Overview

Engineering reference data for Vector Addition in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Methods of Vector Addition

Parallelogram Law

1. Draw vector F₁ to scale in its direction
2. From the tail of F₁, draw vector F₂ to scale in its direction
3. Complete the parallelogram using F₁ and F₂ as sides
4. The resultant F_R is the diagonal from the common tail to the opposite corner

Triangle Rule

1. Draw vector F₁ to scale in its direction
2. From the nose (arrowhead) of F₁, draw vector F₂ to scale in its direction
3. The resultant F_R connects the tail of F₁ to the nose of F₂

Formulas for Trigonometric Calculation

For two vectors F₁ and F₂ with angle θ between them:

Resultant magnitude (using cosine rule): $F_R = \sqrt{F_1^2 + F_2^2 - 2F_1F_2\cos(180° - \theta)}$

Angle between F₁ and resultant (using sine rule): $\alpha = \sin^{-1}\left(\frac{F_1 \sin(180° - \theta)}{F_R}\right)$

Angle between F₂ and resultant: $\beta = \sin^{-1}\left(\frac{F_2 \sin(180° - \theta)}{F_R}\right)$

Example Calculations

Adding Two Forces

• F₁ = 3 kN
• F₂ = 8 kN
• Angle between vectors = 80°

Resultant force: $F_R = \sqrt{(3)^2 + (8)^2 - 2(3)(8)\cos(180° - 80°)} = 9 \text{ kN}$

Angles:

• Between F₁ and F_R: α = 19.1°
• Between F₂ and F_R: β = 60.9°

Airplane in Wind

• Airplane velocity = 900 km/h
• Wind velocity = 100 km/h (headwind, 30° from flight path)
• Angle between vectors = 150° (180° - 30°)

Resultant ground velocity: $v_R = \sqrt{(900)^2 + (100)^2 - 2(900)(100)\cos(150°)} = 815 \text{ km/h}$

Course deviation: α = 3.5°

Properties and Notes

1. Vector addition is commutative: F₁ + F₂ = F₂ + F₁
2. For more than two vectors, add sequentially using parallelogram or triangle method
3. The resultant represents the combined effect of multiple vector quantities
4. The cosine rule formula works for any angle between vectors (not just acute angles)

References

• Original Source