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.
2 · Watch
Take 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.