AI Tutor
If a search plane flies 120Β km120\text{ km} northeast and then turns to fly 80Β km80\text{ km} northwest, how far is it from base camp? It is not 200Β km200\text{ km}, and it is definitely not 40Β km40\text{ km}.
Because direction changes everything, we must treat each leg of the journey as a displacement vectorβ€”a quantity that packs both straight-line distance and direction into a single package.
To solve any navigation puzzle, we need a universal way to turn descriptive compass bearings into precise mathematical angles. How do we translate everyday directions like 'North of West' into math?

Translating Compass Directions

In physics, we measure standard Cartesian angles ΞΈ\theta starting from the positive xx-axis (East) and turning counterclockwise toward North.
Navigational bearings use reference phrases. For instance, 30∘30^\circ North of East means you face East (0∘0^\circ) and swing 30∘30^\circ toward North, giving θ=30∘\theta = 30^\circ.
By contrast, 40∘40^\circ North of West means you face West (180∘180^\circ) and swing 40∘40^\circ toward North, landing at ΞΈ=180βˆ˜βˆ’40∘=140∘\theta = 180^\circ - 40^\circ = 140^\circ.
Once every heading is converted into a standard angle, how do we combine vectors that point in totally different directions?

The Component Method

In 1637, French philosopher RenΓ© Descartes revolutionized navigation by linking algebra with geometry, creating the grid system that lets us break any angled path into independent horizontal and vertical steps.
Because horizontal (xx) and vertical (yy) motions are perpendicular, they do not interfere with each other. We use basic right-triangle trigonometry to find the scalar components of a vector with magnitude dd and standard angle ΞΈ\theta: