Vectors Addition Worksheet PDF Form - FormsPal - Free Printable

Let’s solve this step by step.

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Part I: Find the x and y components of each vector

We use trigonometry:
- x-component = magnitude × cos(angle)
- y-component = magnitude × sin(angle)

Angle is measured from the positive x-axis (standard position).

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First vector: 12 m at 30°

x = 12 × cos(30°) = 12 × (√3/2) ≈ 12 × 0.866 = 10.39 m
y = 12 × sin(30°) = 12 × 0.5 = 6.00 m

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Second vector: 25 m/s at 60°

x = 25 × cos(60°) = 25 × 0.5 = 12.5 m/s
y = 25 × sin(60°) = 25 × (√3/2) ≈ 25 × 0.866 = 21.65 m/s

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Third vector: 15 m/s at 150°

Note: 150° is in second quadrant → x negative, y positive.

x = 15 × cos(150°) = 15 × (-√3/2) ≈ 15 × (-0.866) = -12.99 m/s
y = 15 × sin(150°) = 15 × 0.5 = 7.50 m/s

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Add the following vectors

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First addition: 12 m/s at 20° + 12 m/s at 70°

Break into components:

Vector 1 (20°):
x₁ = 12 × cos(20°) ≈ 12 × 0.9397 = 11.28 m/s
y₁ = 12 × sin(20°) ≈ 12 × 0.3420 = 4.10 m/s

Vector 2 (70°):
x₂ = 12 × cos(70°) ≈ 12 × 0.3420 = 4.10 m/s
y₂ = 12 × sin(70°) ≈ 12 × 0.9397 = 11.28 m/s

Total:
x_tot = 11.28 + 4.10 = 15.38 m/s
y_tot = 4.10 + 11.28 = 15.38 m/s

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Second addition: 50 m at 135° + 30 m at 90°

Vector 1 (135°):
This is in second quadrant.

x₁ = 50 × cos(135°) = 50 × (-√2/2) ≈ 50 × (-0.7071) = -35.36 m
y₁ = 50 × sin(135°) = 50 × (√2/2) ≈ 50 × 0.7071 = 35.36 m

Vector 2 (90°): straight up

x₂ = 30 × cos(90°) = 30 × 0 = 0 m
y₂ = 30 × sin(90°) = 30 × 1 = 30 m

Total:
x_tot = -35.36 + 0 = -35.36 m
y_tot = 35.36 + 30 = 65.36 m

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Final Answer:

Part I:
First vector: x = 10.39 m, y = 6.00 m
Second vector: x = 12.5 m/s, y = 21.65 m/s
Third vector: x = -12.99 m/s, y = 7.50 m/s

Addition 1: x₁ = 11.28 m/s, y₁ = 4.10 m/s; x₂ = 4.10 m/s, y₂ = 11.28 m/s; x_tot = 15.38 m/s, y_tot = 15.38 m/s
Addition 2: x₁ = -35.36 m, y₁ = 35.36 m; x₂ = 0 m, y₂ = 30 m; x_tot = -35.36 m, y_tot = 65.36 m