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Vector components
Beginner
Resolve a vector into perpendicular parts along chosen axes.
Ax = A cosθ, Ay = A sinθAny vector can be split into perpendicular parts along chosen axes: Ax = A cosθ and Ay = A sinθ, where θ is measured from the x-axis. This turns vector problems into simple algebra on each axis.
A 50 N pull at 37° above horizontal has horizontal component 40 N and vertical component 30 N.
Which of sin and cos goes with which axis depends on where θ is measured from. Always check with the limiting cases θ = 0 and 90°.
In more detail
A vector can be split into perpendicular components, usually along the x and y axes. For a vector at angle θ to the x axis, the components are A cos θ and A sin θ.
Real-world uses
- Resolving the weight of a box on a slope into parallel and perpendicular parts to find friction.
- Breaking a projectile's launch velocity into horizontal and vertical components.
- A ramp at 20 degrees splits the weight into a component along the slope and a component into the surface, which is how the engineer sizes a brake.
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Frequently asked questions
How do I find the angle of a vector from its components?
Use the arctangent of the y component divided by the x component, taking the quadrant into account.
Why use components?
They turn a vector problem into two independent one-dimensional problems that are easier to solve.
How do I choose the axes for components?
Pick axes that line up with the motion or the constraint, such as along a slope. This often removes one or more unknown forces from the problem.
Related topics in Measurement & Math Tools
Vector representation · Vector addition · Parallelogram law · Unit vectors · Dot product · Cross product