AI Tutor
When you pull a rolling suitcase through an airport terminal at an upward tilt, only part of your effort propels it forward β€” the rest simply lifts the handle upward.
Splitting a single diagonal vector into two perpendicular parts along coordinate axes is called orthogonal resolution, where orthogonal means meeting at a right angle (90∘90^\circ).
How do we calculate the exact numerical lengths of these perpendicular pieces from just the magnitude and direction of the main vector?

Trigonometric Resolution

Any 2D vector Aβƒ—\vec{A} forms the hypotenuse (the longest side across from the 90∘90^\circ angle) of a right triangle whose legs are the horizontal component AxA_x and vertical component AyA_y.
In the 2nd century BCE, Greek astronomer Hipparchus tabulated geometric chords to track planetary positions, laying the groundwork for right-triangle ratios: cosine relates the adjacent side to the hypotenuse, while sine relates the opposite side.
Using standard definitions where ΞΈ\theta is measured counterclockwise from the positive xx-axis:
cos⁑θ=adjacenthypotenuse=AxAβ€…β€ŠβŸΉβ€…β€ŠAx=Acos⁑θ\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{A_x}{A} \implies A_x = A\cos\theta
sin⁑θ=oppositehypotenuse=AyAβ€…β€ŠβŸΉβ€…β€ŠAy=Asin⁑θ\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{A_y}{A} \implies A_y = A\sin\theta
What does this resolution process look like when we plug in actual physical forces?

Worked Example: Pulling a Sled