For AI agents: use /llms.txt for the Nakafa content index.
When a vector forms an angle with an axis, its components can be found using trigonometry. The idea is simple: the slanted vector becomes the hypotenuse of a right triangle, while its components become the horizontal and vertical sides.
If angle is measured from the positive axis, the horizontal component is adjacent to the angle, while the vertical component is opposite the angle.
Do not memorize where and go without checking the reference angle.
Imagine a rope pulling a box with force up and to the right. The pull has a horizontal part that pulls the box rightward and a vertical part that lifts slightly upward.
The endpoint of followed by matches the endpoint of vector . The components are not extra forces. They are another way to describe the same force.
A force acts above the horizontal. Use and .
So the slanted force is equivalent to a pull to the right and a pull upward.
The two components are not extra forces added to the situation. They are another way to describe the same force using horizontal and vertical directions.
The formulas change when the reference angle changes. If angle is measured from the axis, the component adjacent to the angle is .
Before using a formula, always ask: from which axis is the angle measured? This small check prevents a common error when resolving vectors.
The values of and give component sizes, but positive or negative signs come from the vector direction on the coordinate axes.
| Vector direction | Sign of | Sign of |
|---|---|---|
| up and right | positive | positive |
| up and left | negative | positive |
| down and left | negative | negative |
| down and right | positive | negative |
If a force points up and to the left, its horizontal component is negative because it opposes the positive axis, while its vertical component remains positive.
