Vector addition worksheet with problems involving finding components and adding vectors.

Absolutely! Let’s solve this vector addition worksheet step by step. We’ll break it into two parts:

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

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

Where θ is measured counterclockwise from the positive x-axis.

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Vector 1: 12 m at 30°

This vector is in the first quadrant, 30° above the x-axis.

- x = 12 × cos(30°) = 12 × (√3/2) ≈ 12 × 0.8660 = 10.392 m
- y = 12 × sin(30°) = 12 × 0.5 = 6.000 m

✔ x = 10.39 m, y = 6.00 m (rounded to 2 decimals)

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Vector 2: 25 m/s at 60°

Also in first quadrant.

- x = 25 × cos(60°) = 25 × 0.5 = 12.50 m/s
- y = 25 × sin(60°) = 25 × (√3/2) ≈ 25 × 0.8660 = 21.65 m/s

✔ x = 12.50 m/s, y = 21.65 m/s

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Vector 3: 15 m/s at 150°

This is in the second quadrant (180° - 30°).

- x = 15 × cos(150°) = 15 × (-cos(30°)) = 15 × (-√3/2) ≈ 15 × (-0.8660) = -12.99 m/s
- y = 15 × sin(150°) = 15 × sin(30°) = 15 × 0.5 = 7.50 m/s

✔ x = -12.99 m/s, y = 7.50 m/s

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## Part II: Add the following vectors

We’ll find components for each vector, then add x-components together and y-components together.

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Problem 1: Add 12 m/s @ 20° + 12 m/s @ 70°

#### First vector: 12 m/s @ 20°

- x₁ = 12 × cos(20°) ≈ 12 × 0.9397 = 11.276 m/s
- y₁ = 12 × sin(20°) ≈ 12 × 0.3420 = 4.104 m/s

#### Second vector: 12 m/s @ 70°

- x₂ = 12 × cos(70°) ≈ 12 × 0.3420 = 4.104 m/s
- y₂ = 12 × sin(70°) ≈ 12 × 0.9397 = 11.276 m/s

#### Total:

- x_tot = x₁ + x₂ = 11.276 + 4.104 = 15.380 m/s
- y_tot = y₁ + y₂ = 4.104 + 11.276 = 15.380 m/s

✔ 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

*(Note: You may notice that since both vectors have same magnitude and are symmetric about 45°, their sum should be along 45° with magnitude 12×√2 ≈ 16.97, but component-wise we’re just adding — which is correct.)*

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Problem 2: Add 50 m @ 135° + 30 m @ 90°

#### First vector: 50 m @ 135°

This is in second quadrant.

- x₁ = 50 × cos(135°) = 50 × (-cos(45°)) = 50 × (-√2/2) ≈ 50 × (-0.7071) = -35.355 m
- y₁ = 50 × sin(135°) = 50 × sin(45°) = 50 × (√2/2) ≈ 50 × 0.7071 = 35.355 m

#### Second vector: 30 m @ 90°

Straight up the y-axis.

- x₂ = 30 × cos(90°) = 30 × 0 = 0.000 m
- y₂ = 30 × sin(90°) = 30 × 1 = 30.000 m

#### Total:

- x_tot = x₁ + x₂ = -35.355 + 0 = -35.355 m
- y_tot = y₁ + y₂ = 35.355 + 30 = 65.355 m

✔ x₁ = -35.36 m, y₁ = 35.36 m
✔ x₂ = 0.00 m, y₂ = 30.00 m
✔ x_tot = -35.36 m, y_tot = 65.36 m

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## ✔ Final Answers Summary:

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Part I: Components

1. 12 m @ 30° → x = 10.39 m, y = 6.00 m
2. 25 m/s @ 60° → x = 12.50 m/s, y = 21.65 m/s
3. 15 m/s @ 150° → x = -12.99 m/s, y = 7.50 m/s

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Part II: Vector Addition

#### Problem 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

#### Problem 2:
- x₁ = -35.36 m, y₁ = 35.36 m
- x₂ = 0.00 m, y₂ = 30.00 m
- x_tot = -35.36 m, y_tot = 65.36 m

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Let me know if you’d like to also find the magnitude and direction of the resultant vectors — that’s often the next step!