Engineering question: How can we describe an angled force using horizontal and vertical components—and combine several forces into one resultant?
Complete the modules in order. The diagrams and video prompts are designed for learning without a separate teacher presentation.
For independent pacing, you can use the optional short video and additional practice after class.
A scalar has magnitude only (such as 100 kg of mass). A vector has magnitude and direction (such as a 100 N pull at 30° upward and to the right).
Imagine a cable pulling a joint up and right. Do you expect both x and y components to be positive? What if the cable pulls down and right instead?
Up/right gives +x, +y. Down/right gives +x, −y. The force magnitude stays positive; component signs tell us the direction.
We measure θ counterclockwise from the positive x-axis, unless a problem explicitly says otherwise. A clockwise angle can be written as negative.
Calculator: Set the trig mode to DEG (degrees), not radians, for all problems on this page.
To resolve means to replace one vector with horizontal and vertical vectors that add back to the original force.
Fₓ lies along the x-axis. For the drawn 30° force, cosine calculates the horizontal share.
Fₓ = 100 cos 30° ≈ +86.60 N
Fᵧ lies along the y-axis. For the drawn 30° force, sine calculates the vertical share.
Fᵧ = 100 sin 30° = +50.00 N
The same positive force magnitude can generate positive or negative components depending on which quadrant contains the vector.
Standard-angle convention: 0° points right, 90° points up, 180° points left, and −90° points down.
| Angle θ | Fₓ | Fᵧ |
|---|---|---|
| +30° | Positive | Positive |
| +150° | Negative | Positive |
| −135° | Negative | Negative |
| −60° | Positive | Negative |
Worked sign check: For 50 N at −60°, Fₓ = 50 cos(−60°) = +25.0 N; Fᵧ = 50 sin(−60°) = −43.30 N.
Begin with vector basics, then study analytical vector addition. The short third video is helpful if you only need a quick review of resolving one force.
Pause and practice: Write the difference between a scalar and vector, then draw one force with a positive x component and negative y component.
Open on YouTube ↗Pause and practice: Pause before the resultant is calculated. Try adding x components and y components independently.
Open on YouTube ↗Pause and practice: Predict which leg is adjacent to the stated angle and label the signs.
Open on YouTube ↗Follow the same method every time: sketch axes → identify θ → resolve → assign signs → check the result.
F = (86.60 i + 50.00 j) N
| Force | Fₓ (N) | Fᵧ (N) |
|---|---|---|
| F₁ = 80 N @ +45° | +56.57 | +56.57 |
| F₂ = 50 N @ −60° | +25.00 | −43.30 |
| Sum | +81.57 | +13.27 |
Magnitude: R = √(81.57² + 13.27²) = 82.64 N.
Direction: θ = atan2(+13.27, +81.57) ≈ +9.24° above +x.
Intermediate values are rounded for display. Calculations use unrounded component values.
Add only x components together; add only y components together. Never add magnitudes directly unless the vectors point along the same line in the same direction.
Calculate direction from the component signs and a quadrant-aware inverse tangent: θ = atan2(Rᵧ,Rₓ). If your calculator only has tan⁻¹, sketch the quadrant and correct the angle accordingly.
Adjust magnitude and angle of Force 1. Then activate a second force (50 N at −60°) to see their component sum and resultant. The graph and numbers update immediately.
In Quadrant I: +x, +y. Horizontal component is positive; vertical component is positive.
A diagonal truss member acts along its own axis. The force it exerts on a joint can be understood as an x force and a y force. This helps solve joint equilibrium in later Unit 2 lessons.
This diagram shows an example of a diagonal tension member pulling a lower joint up and right. A member in compression would exert a different force direction on that joint.
Do not confuse components with extra forces: The x and y arrows are an equivalent representation of the original diagonal force. Do not count all three when summing forces.
Sketch axes and a force arrow for each problem. Use DEG mode, show trig setup, use signs and round final values to two decimals.
Find the signed horizontal and vertical components.
Fₓ = 75 cos60° = +37.50 N. Fᵧ = 75 sin60° = +64.95 N. Both are positive in quadrant I.
Calculate Fₓ and Fᵧ. Which component is negative?
Fₓ = 90 cos(−30°) = +77.94 N; Fᵧ = 90 sin(−30°) = −45.00 N. The force points down/right.
Find both components and verify their signs from a sketch.
Fₓ = 40 cos150° = −34.64 N; Fᵧ = 40 sin150° = +20.00 N. The force points left/up.
Find magnitude and direction as a counterclockwise angle from +x.
Magnitude = √(30² + 40²) = 50.00 N. Vector is in quadrant II: θ = atan2(+40, −30) ≈ +126.87°.
Calculate the resultant components, magnitude and direction.
Rₓ = 56.57 + 25.00 = +81.57 N; Rᵧ = 56.57 − 43.30 = +13.27 N. |R| ≈ 82.64 N, θ ≈ +9.24°.
Explain whether sine or cosine gives the horizontal component. Then calculate both components.
The angle is measured from +y, so horizontal is opposite: Fₓ = 100 sin20° = +34.20 N. Vertical is adjacent: Fᵧ = 100 cos20° = +93.97 N.
These open textbooks expand the explanations above and include additional examples and diagrams.
Open engineering textbook covering orthogonal components, coordinate axes, force addition and an interactive vector diagram.
Read Engineering Statics Section 2.6 ↗Free university physics reading with diagrams and equations for vector coordinates and components.
Read OpenStax: Coordinate Systems ↗Accessible force-components and resultant-force examples, from diagrams to calculated magnitude.
Read OpenStax: Analytical Addition ↗Resultant: a single vector with the same combined effect as several vectors.
Component: one perpendicular part of a force vector.
Quadrant: one of four regions of the xy-plane.
Resolve: express a vector as two perpendicular components.
Complete your POE 2.5 Calculating Force Vectors classwork and homework worksheet (or equivalent teacher-directed notebook work). The examples and revealed solutions are for learning and checking—not a substitute for showing your own work.
Due: Check the Daily Agenda for the current deadline. Follow the teacher's submission directions.
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