Vector Addition Physics Worksheet - Free Printable

We are given 5 right triangles representing vector addition problems. In each, we must compute the missing side(s) and/or angle using trigonometry (Pythagorean Theorem, sine, cosine, tangent).

Let’s solve each one step by step.

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Problem 1:
Given:
a = 20 m (adjacent to angle α)
b = 15 m (opposite to angle α)
Right angle between a and b → c is hypotenuse

Find: c and α

- Use Pythagorean Theorem for c:
c = √(a² + b²) = √(20² + 15²) = √(400 + 225) = √625 = 25 m

- Use tangent for angle α:
tan(α) = opposite / adjacent = b/a = 15/20 = 0.75
α = arctan(0.75) ≈ 36.87°

✔ Answer 1: c = 25 m, α ≈ 36.87°

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Problem 2:
Given:
c = 5.0 m/s (hypotenuse)
β = 35° (angle at bottom left, between side a and hypotenuse c)
Right angle between a and b

Find: a and b

- a is adjacent to β → use cosine:
cos(β) = a/c → a = c * cos(β) = 5.0 * cos(35°) ≈ 5.0 * 0.8192 ≈ 4.10 m/s

- b is opposite to β → use sine:
sin(β) = b/c → b = c * sin(β) = 5.0 * sin(35°) ≈ 5.0 * 0.5736 ≈ 2.87 m/s

✔ Answer 2: a ≈ 4.10 m/s, b ≈ 2.87 m/s

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Problem 3:
Given:
a = 10.0 m (adjacent to θ)
c = 12.5 m (hypotenuse)

Find: b and θ

- Use Pythagorean Theorem for b:
b = √(c² - a²) = √(12.5² - 10.0²) = √(156.25 - 100) = √56.25 = 7.5 m

- Use cosine for θ:
cos(θ) = a/c = 10.0 / 12.5 = 0.8
θ = arccos(0.8) ≈ 36.87°

✔ Answer 3: b = 7.5 m, θ ≈ 36.87°

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Problem 4:
Given:
a = 36 m/s (horizontal leg)
b = 25 m/s (vertical leg)
Right angle between them → c is hypotenuse

Find: c and θ

- Use Pythagorean Theorem for c:
c = √(a² + b²) = √(36² + 25²) = √(1296 + 625) = √1921 ≈ 43.83 m/s

- Use tangent for θ (angle at top right, between a and c):
tan(θ) = opposite / adjacent = b/a = 25/36 ≈ 0.6944
θ = arctan(0.6944) ≈ 34.8°

✔ Answer 4: c ≈ 43.83 m/s, θ ≈ 34.8°

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Problem 5:
Given:
c = 45 m (hypotenuse)
α = 25° (angle at bottom right, between side a and hypotenuse c)
Right angle between a and b

Find: a and b

- a is adjacent to α → use cosine:
cos(α) = a/c → a = c * cos(α) = 45 * cos(25°) ≈ 45 * 0.9063 ≈ 40.78 m

- b is opposite to α → use sine:
sin(α) = b/c → b = c * sin(α) = 45 * sin(25°) ≈ 45 * 0.4226 ≈ 19.02 m

✔ Answer 5: a ≈ 40.78 m, b ≈ 19.02 m

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## ✔ Final Answers:

1. c = 25 m, α ≈ 36.87°
2. a ≈ 4.10 m/s, b ≈ 2.87 m/s
3. b = 7.5 m, θ ≈ 36.87°
4. c ≈ 43.83 m/s, θ ≈ 34.8°
5. a ≈ 40.78 m, b ≈ 19.02 m

All values are rounded to two decimal places unless exact (like 25, 7.5). Angles are in degrees.

These solutions use basic right-triangle trigonometry — SOHCAHTOA and Pythagoras — which are standard for vector component calculations.