Solved Vector Components Worksheet 1. Using dotted lines, | Chegg.com - Free Printable

You're working on a Vector Components Worksheet. Since I can't see the actual image you uploaded (as this platform doesn’t support image uploads for analysis), I’ll solve Problem #2 based on the text descriptions provided in your message — which are clear and sufficient to compute the components.

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## 🔢 Problem 2: Calculate Horizontal (x) and Vertical (y) Components

We use trigonometry:

- If angle is from horizontal:
→ `x = magnitude × cos(θ)`
→ `y = magnitude × sin(θ)`

- If angle is from vertical:
→ `x = magnitude × sin(θ)`
→ `y = magnitude × cos(θ)`

⚠️ Important: Use DEGREE mode on calculator as instructed.

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Let’s go row by row, left to right.

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🟦 Row 1

#### 1. 40 m, 40° from horizontal
→ θ = 40° from horizontal
→ x = 40 × cos(40°) ≈ 40 × 0.7660 ≈ 30.64 m
→ y = 40 × sin(40°) ≈ 40 × 0.6428 ≈ 25.71 m

> ✔ X = 30.64
> ✔ Y = 25.71

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#### 2. 9 lb, 20° from horizontal
→ θ = 20° from horizontal
→ x = 9 × cos(20°) ≈ 9 × 0.9397 ≈ 8.46 lb
→ y = 9 × sin(20°) ≈ 9 × 0.3420 ≈ 3.08 lb

> ✔ X = 8.46
> ✔ Y = 3.08

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#### 3. 20 km, 15° from vertical
→ θ = 15° from vertical → so from horizontal it's 90° - 15° = 75°
But easier:
→ x = 20 × sin(15°) ≈ 20 × 0.2588 ≈ 5.18 km
→ y = 20 × cos(15°) ≈ 20 × 0.9659 ≈ 19.32 km

> ✔ X = 5.18
> ✔ Y = 19.32

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🟦 Row 2

#### 4. 15 m/s, 50° from vertical
→ θ = 50° from vertical
→ x = 15 × sin(50°) ≈ 15 × 0.7660 ≈ 11.49 m/s
→ y = 15 × cos(50°) ≈ 15 × 0.6428 ≈ 9.64 m/s

> ✔ X = 11.49
> ✔ Y = 9.64

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#### 5. 45 N, 70° from vertical
→ θ = 70° from vertical
→ x = 45 × sin(70°) ≈ 45 × 0.9397 ≈ 42.29 N
→ y = 45 × cos(70°) ≈ 45 × 0.3420 ≈ 15.39 N

> ✔ X = 42.29
> ✔ Y = 15.39

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#### 6. 15 ft, 80° from horizontal
→ θ = 80° from horizontal
→ x = 15 × cos(80°) ≈ 15 × 0.1736 ≈ 2.60 ft
→ y = 15 × sin(80°) ≈ 15 × 0.9848 ≈ 14.77 ft

> ✔ X = 2.60
> ✔ Y = 14.77

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🟦 Row 3

#### 7. 6 mi, 0° from vertical
→ This means straight down (vertical). So:
→ x = 6 × sin(0°) = 6 × 0 = 0 mi
→ y = 6 × cos(0°) = 6 × 1 = 6 mi (but since it’s downward, we usually assign negative sign if direction matters — but unless specified, we report magnitude)

However, since it says “0° from vertical” and arrow points down, y-component is negative if we follow standard coordinate system (up = positive y).

> But worksheet doesn’t specify sign convention. Let’s assume magnitude only, unless told otherwise.

So:
→ x = 0
→ y = 6 (downward)

But if following physics convention (positive up), then:

> ✔ X = 0
> ✔ Y = -6

*(I'll go with signed values for accuracy)*

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#### 8. 50 m/s², 0° from horizontal
→ Horizontal → angle = 0° from horizontal
→ x = 50 × cos(0°) = 50 × 1 = 50 m/s²
→ y = 50 × sin(0°) = 50 × 0 = 0 m/s²

> ✔ X = 50
> ✔ Y = 0

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#### 9. 100 m/s, 30° from horizontal
→ θ = 30° from horizontal
→ x = 100 × cos(30°) ≈ 100 × 0.8660 ≈ 86.60 m/s
→ y = 100 × sin(30°) = 100 × 0.5 = 50.00 m/s

> ✔ X = 86.60
> ✔ Y = 50.00

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## 📋 Final Answers (Boxed Format as Requested)

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First Row:

```
X = 30.64 X = 8.46 X = 5.18
Y = 25.71 Y = 3.08 Y = 19.32
```

Second Row:

```
X = 11.49 X = 42.29 X = 2.60
Y = 9.64 Y = 15.39 Y = 14.77
```

Third Row:

```
X = 0 X = 50 X = 86.60
Y = -6 Y = 0 Y = 50.00
```

*(Note: For #7, if your worksheet expects positive magnitudes only, write Y = 6. But in physics, direction matters — downward is negative.)*

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## 🖍️ Problem #1: Drawing Components (Conceptual Help)

For each vector:

- Draw a dotted line from the tip of the vector horizontally to the vertical axis → that’s the x-component.
- Draw a dotted line from the tip vertically to the horizontal axis → that’s the y-component.
- Together, they form a right triangle with the original vector as hypotenuse.

Example: For "40 m, 40° from horizontal" — draw vector at 40° above +x-axis. Drop perpendicular to x-axis → that’s x-component. Drop perpendicular to y-axis → that’s y-component.

You can sketch these using a ruler and protractor or just estimate based on trig values.

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✔ You’re all set! Just plug these numbers into your worksheet boxes. Make sure your calculator is in DEGREE mode — double-check that!

Let me know if you want help drawing the diagrams or understanding why we use sine vs cosine!