--- title: "Forces as Vectors: Resolving and Resultants" description: "Draw every force acting on a body as an arrow, add them as vectors to get the resultant, and split any force into components along two perpendicular directions of your choosing." canonical: https://lightmysky.com/learn/mathematics/forces-as-vectors-resolving-and-resultants-mt_-qDRgd2g8v source: https://lightmysky.com/learn/mathematics/forces-as-vectors-resolving-and-resultants-mt_-qDRgd2g8v.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Forces as Vectors: Resolving and Resultants Draw every force acting on a body as an arrow, add them as vectors to get the resultant, and split any force into components along two perpendicular directions of your choosing. Subject: Mathematics · Area: Mechanics · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/forces-as-vectors-resolving-and-resultants-mt_-qDRgd2g8v ## Ready when they can - Resolve a 20 N force at 30° to the horizontal into two components - Find the magnitude and direction of the resultant of two perpendicular forces - Draw a diagram showing every force on a body and nothing that is not a force ## Lesson: Splitting and adding forces A tilted force is two forces in disguise: one across and one up, at right angles to each other. With the angle measured from the horizontal, the across part is F times cos and the up part is F times sin. A 20 N force at 30 degrees gives 20 times 0.5, which is 10 N upward, and about 17.3 N across. Perpendicular forces add through Pythagoras, and the direction comes from tan. A 3 N pull east with a 4 N pull north gives root 25, which is 5 N, pointing about 53 degrees north of east. A 6 N pull east with an 8 N pull north gives root 100, which is 10 N. **Example.** A free body diagram shows every force on the body and nothing else. For a book resting on a table, draw gravity down and the normal force up, with each arrow starting on the book. The push of the book on the table acts on the table, so it stays out of the book diagram. **Tip.** Split tilted forces into parts before you balance anything. Each direction then balances on its own, which turns one tilted problem into two straight line ones. **Recap.** Split each force into across and up parts, add perpendicular forces with Pythagoras, and draw only forces on the body. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Kinematics with Calculus](https://lightmysky.com/learn/mathematics/kinematics-with-calculus-mt_2kNspOOoDw) - [Trigonometry basics](https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH) - [Vectors: Notation, Addition and Scalar Multiples](https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe) - [Bearing Problems with the Sine and Cosine Rules](https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP) ## Opens up - [Equilibrium and Newton's First Law](https://lightmysky.com/learn/mathematics/equilibrium-and-newtons-first-law-mt_e6RFRQkRPo) - [Friction and the Coefficient of Friction](https://lightmysky.com/learn/mathematics/friction-and-the-coefficient-of-friction-mt_EwFUd9N24W) - [The Dot Product: Angles and Projections](https://lightmysky.com/learn/mathematics/the-dot-product-angles-and-projections-mt_yFBcyE4sFw) - [Newton's Second Law in Two Dimensions](https://lightmysky.com/learn/mathematics/newtons-second-law-in-two-dimensions-mt_yNUFF3lxlj)