Vector Components Worksheet 1. Using dotted lines, draw the - Free Printable

To solve this problem, we need to calculate the horizontal (x) and vertical (y) components of each vector using trigonometry.

The formulas are:
- Horizontal component (X) = magnitude × cos(angle)
- Vertical component (Y) = magnitude × sin(angle)

Important: The angle is measured from either the horizontal or vertical axis as specified. We must be careful about which axis the angle is referenced to, and also the direction (positive or negative) based on the vector’s orientation.

Let’s go one by one.

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1. 40 m, 40° from horizontal

This means the angle is measured from the horizontal axis. Since the arrow points up and left, it’s in the second quadrant → X will be negative, Y positive.

But for component calculation, we use the reference angle from the horizontal.

X = 40 × cos(40°) = 40 × 0.7660 ≈ 30.64 → but since it's pointing left, X = -30.64 m
Y = 40 × sin(40°) = 40 × 0.6428 ≈ 25.71 m

Wait — actually, if the angle is “from horizontal” and the vector is going up-left, then yes, X is negative.

But let’s check the diagram description: the first box says “40 m, 40° from horizontal” and shows an arrow pointing up and to the left. So yes, angle is 40° above the negative x-axis? Or is it 40° from the positive horizontal?

Actually, standard convention: when it says “θ from horizontal”, it usually means from the positive x-axis unless otherwise specified. But here, since the arrow is drawn going up-left, and labeled “40° from horizontal”, it likely means 40° above the negative x-axis? That would make the actual angle from positive x-axis = 180° - 40° = 140°.

But to avoid confusion, let’s interpret based on common worksheet practice:

In most such worksheets, when they say “θ from horizontal” and show a vector going up-left, they mean the acute angle between the vector and the horizontal line — so we take the magnitude and apply trig functions, then assign signs based on direction.

So for “40 m, 40° from horizontal” with arrow up-left:

→ X = -40 × cos(40°) ≈ -30.64 m
→ Y = +40 × sin(40°) ≈ +25.71 m

Similarly, for others.

Let me tabulate all 9 vectors carefully.

I’ll list them in order as they appear in the grid (row by row):

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Row 1:

A. 40 m, 40° from horizontal → arrow up-left
Angle from horizontal = 40°, so:
X = -40 * cos(40°) = -40 * 0.7660 = -30.64 m
Y = +40 * sin(40°) = 40 * 0.6428 = +25.71 m

B. 9 lb, 20° from horizontal → arrow up-right
X = +9 * cos(20°) = 9 * 0.9397 = +8.46 lb
Y = +9 * sin(20°) = 9 * 0.3420 = +3.08 lb

C. 20 km, 15° from vertical → arrow down-right? Wait, diagram shows arrow pointing down and right? Actually, looking at description: “20 km 15° from vertical” — and arrow is drawn going down and slightly right? Or up? Let me think.

Actually, in the original image description, it says: “20 km 15° from vertical” and the arrow is shown going downward and to the right? But typically, if it’s “from vertical”, and arrow is going down-right, then angle from vertical is 15°, so from horizontal it would be 75°.

But to be precise:

If angle is from vertical, then:

Horizontal component = magnitude * sin(angle from vertical)
Vertical component = magnitude * cos(angle from vertical)

And sign depends on direction.

Assuming the arrow is pointing down and to the right (as commonly drawn for such problems):

→ X = +20 * sin(15°) = 20 * 0.2588 = +5.18 km
→ Y = -20 * cos(15°) = -20 * 0.9659 = -19.32 km

But wait — let’s confirm the direction. In many worksheets, if not specified, we assume the angle is measured from the positive axis, but here it says “from vertical”. To avoid error, let’s stick to:

When angle is given “from vertical”, then:

- The component along the vertical is magnitude * cos(angle)
- The component along horizontal is magnitude * sin(angle)

Signs depend on direction.

For “20 km, 15° from vertical” — if the arrow is pointing down and to the right, then:

X = +20 * sin(15°) ≈ +5.18 km
Y = -20 * cos(15°) ≈ -19.32 km

But let’s look at the next ones to see pattern.

Actually, I recall that in some worksheets, “from vertical” means the angle between the vector and the vertical line, so we use sin for horizontal, cos for vertical.

Yes.

So continuing.

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Row 2:

D. 15 m/s, 50° from vertical → arrow down-right? Diagram shows arrow going down and to the right? Assume so.

Angle from vertical = 50°, so:

X = +15 * sin(50°) = 15 * 0.7660 = +11.49 m/s
Y = -15 * cos(50°) = -15 * 0.6428 = -9.64 m/s

E. 45 N, 70° from vertical → arrow up-left? Diagram shows arrow going up and to the left? Assume so.

Angle from vertical = 70°, so:

X = -45 * sin(70°) = -45 * 0.9397 = -42.29 N
Y = +45 * cos(70°) = 45 * 0.3420 = +15.39 N

F. 15 ft, 80° from horizontal → arrow up-right? Diagram shows arrow going up and to the right? Assume so.

Angle from horizontal = 80°, so:

X = +15 * cos(80°) = 15 * 0.1736 = +2.60 ft
Y = +15 * sin(80°) = 15 * 0.9848 = +14.77 ft

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Row 3:

G. 6 mi, 0° from vertical → arrow straight down? Diagram shows vertical arrow down.

0° from vertical means it’s aligned with vertical. Since it’s pointing down:

X = 6 * sin(0°) = 0
Y = -6 * cos(0°) = -6 * 1 = -6 mi

H. 50 m/s², 0° from horizontal → arrow straight right.

0° from horizontal means along positive x-axis.

X = 50 * cos(0°) = 50 * 1 = +50 m/s²
Y = 50 * sin(0°) = 0

I. 100 m/s, 30° from horizontal → arrow down-right? Diagram shows arrow going down and to the right.

So angle from horizontal is 30° below horizontal.

Thus:

X = +100 * cos(30°) = 100 * 0.8660 = +86.60 m/s
Y = -100 * sin(30°) = -100 * 0.5 = -50.00 m/s

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Now, let’s compile all answers in order as per the worksheet boxes.

The worksheet has 9 boxes in 3 rows, and below, 9 answer boxes in same order.

So:

Box 1 (top-left): 40 m, 40° from horizontal → X = -30.64, Y = 25.71
Box 2 (top-middle): 9 lb, 20° from horizontal → X = 8.46, Y = 3.08
Box 3 (top-right): 20 km, 15° from vertical → X = 5.18, Y = -19.32
Box 4 (middle-left): 15 m/s, 50° from vertical → X = 11.49, Y = -9.64
Box 5 (middle-middle): 45 N, 70° from vertical → X = -42.29, Y = 15.39
Box 6 (middle-right): 15 ft, 80° from horizontal → X = 2.60, Y = 14.77
Box 7 (bottom-left): 6 mi, 0° from vertical → X = 0, Y = -6.00
Box 8 (bottom-middle): 50 m/s², 0° from horizontal → X = 50.00, Y = 0
Box 9 (bottom-right): 100 m/s, 30° from horizontal → X = 86.60, Y = -50.00

Let me double-check calculations with calculator values.

Recalculate with more precision:

cos(40°) = 0.7660444431 → 40*0.7660444431 = 30.64177772 → -30.64
sin(40°) = 0.6427876097 → 40*0.6427876097 = 25.71150439 → 25.71

cos(20°) = 0.9396926208 → 9*0.9396926208 = 8.457233587 → 8.46
sin(20°) = 0.3420201433 → 9*0.3420201433 = 3.07818129 → 3.08

sin(15°) = 0.2588190451 → 20*0.2588190451 = 5.176380902 → 5.18
cos(15°) = 0.9659258263 → 20*0.9659258263 = 19.31851653 → 19.32 → Y = -19.32

sin(50°) = 0.7660444431 → 15*0.7660444431 = 11.49066665 → 11.49
cos(50°) = 0.6427876097 → 15*0.6427876097 = 9.641814145 → 9.64 → Y = -9.64

sin(70°) = 0.9396926208 → 45*0.9396926208 = 42.28616794 → 42.29 → X = -42.29
cos(70°) = 0.3420201433 → 45*0.3420201433 = 15.39090645 → 15.39

cos(80°) = 0.1736481777 → 15*0.1736481777 = 2.604722665 → 2.60
sin(80°) = 0.984807753 → 15*0.984807753 = 14.77211629 → 14.77

sin(0°) = 0 → X=0
cos(0°)=1 → Y=-6*1 = -6.00

cos(0°)=1 → X=50*1=50.00
sin(0°)=0 → Y=0

cos(30°)=√3/2≈0.8660254038 → 100*0.8660254038=86.60254038 → 86.60
sin(30°)=0.5 → 100*0.5=50 → Y=-50.00

All good.

Now, for the final answer, we need to fill in the boxes as per the worksheet.

The worksheet has 9 answer boxes, each with X= and Y=.

So, in order:

1. X = -30.64, Y = 25.71
2. X = 8.46, Y = 3.08
3. X = 5.18, Y = -19.32
4. X = 11.49, Y = -9.64
5. X = -42.29, Y = 15.39
6. X = 2.60, Y = 14.77
7. X = 0.00, Y = -6.00
8. X = 50.00, Y = 0.00
9. X = 86.60, Y = -50.00

I think that’s it.

Final Answer:

X = -30.64, Y = 25.71
X = 8.46, Y = 3.08
X = 5.18, Y = -19.32
X = 11.49, Y = -9.64
X = -42.29, Y = 15.39
X = 2.60, Y = 14.77
X = 0.00, Y = -6.00
X = 50.00, Y = 0.00
X = 86.60, Y = -50.00