Resolving Vectors into Components
Splitting one vector into two perpendicular parts with sine and cosine, then adding several vectors by adding their components. Scale drawing gives way to a method that stays exact however many forces act.
What a learner can do afterwards
- Resolves a force or a velocity into components along any chosen pair of perpendicular axes
- Adds three or more coplanar vectors by components and states the resultant as a size and an angle
- Picks the axes that make a problem shortest and says why that choice is free
1 · Read
A kicked ball flies sideways and upward from a single push. Resolving splits that one vector into perpendicular parts, usually horizontal and vertical, where each part acts alone along its axis. From size A and angle from the x axis, the x part is A times cos and the y part is A times sin, since cosine sits beside the angle. A velocity of 20 metres per second at 30 degrees gives 20 times 0.5, which is 10 upward, and about 17.3 sideways.
Adding vectors by components is bookkeeping: add all x parts for the total x, add all y parts for the total y, then size and angle come from Pythagoras and tan. Three pulls on a ring with x parts 4, 2 and 2 add to 8, and y parts 1, 2 and 3 add to 6, so the resultant is 10 at about 37 degrees above east.
Pick the axes that kill the most terms: align one axis with the motion or the slope to shorten the maths. On a slope of angle theta, weight splits into m g sin theta down the slope and m g cos theta into it. Tilt swaps which trig sits where, so always sketch the triangle before you write. The axes are your measuring choice: turning them changes the numbers but not the motion itself.
You may point the axes anywhere: the physics is unchanged, only the arithmetic gets easier or harder. If your angle is measured from the y axis instead, sine and cosine swap sides.
Split each vector with cos beside the angle, add part by part, and choose axes that kill terms.
The rewritten reading was written for this dataset.
More to read
Describe a vector along two axes
Draw horizontal and vertical axes that meet at the origin. Put the arrow's tail there, then draw perpendicular lines from its tip to the axes to read the components. Choose right and up as positive; left and down then give negative values. A sloping push or pull can be represented by horizontal and vertical components. This turns a two-dimensional problem into two linked one-dimensional problems.
2.04: Coordinate Systems and Components of a Vector (Part 1)
Coordinate systems are scaffolding, not scenery. You plant the origin where the action starts and point the axes where the maths stays short. A vector keeps its length and direction whatever frame you pick; only its component numbers change. Switching frames never changes the physics, which is why the choice is free. Practise replanting axes until the numbers fall out with least effort.
2.05: Coordinate Systems and Components of a Vector (Part 2)
Adding three or more vectors is just bookkeeping once components exist. Total every x part into one sum and every y part into another, minding the signs. The resultant stretches from the origin to that total point, with size from Pythagoras and direction from the tangent. State both size and angle or the answer is half done. Neat columns of components beat head to tail sketches for anything past two arrows.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.