AI Tutor
Imagine a camera drone tracking an athlete, flying diagonally across a field at 37∘37^\circ north of east at a speed of 10 m/s10\text{ m/s}.
How do we figure out how fast it moves purely eastward versus purely northward?
To solve this, we resolve vectors into orthogonal Cartesian components — breaking two-dimensional motion into independent horizontal (xx) and vertical (yy) parts.
How do we describe an object's position on a flat plane before it even starts moving?

Position and Displacement Vectors

A position vector r⃗\vec{r} points directly from the origin (0,0)(0,0) to the object's coordinates (x,y)(x,y), written as r⃗=xi^+yj^\vec{r} = x\hat{i} + y\hat{j}.
The symbols i^\hat{i} and j^\hat{j} are unit vectors — directional markers with a magnitude of exactly 11 pointing along the positive xx and yy axes.
When an object shifts from initial position r⃗1\vec{r}_1 to final position r⃗2\vec{r}_2, its displacement Δr⃗\Delta \vec{r} is the straight-line difference Δr⃗=r⃗2−r⃗1=(x2−x1)i^+(y2−y1)j^\Delta \vec{r} = \vec{r}_2 - \vec{r}_1 = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j}.
If we only know a vector's overall magnitude and launch angle, how do we split it into coordinates — and how do we rebuild it later?

Resolving and Reconstructing Vectors

In 1637, René Descartes introduced coordinate geometry to connect algebra and shapes, giving us the right-triangle tools we use to resolve vectors.