AI Tutor
If a delivery drone lifts off and flies across a city, a single number like "500 meters" tells you almost nothing about where it actually ended up. To locate an object in a flat plane, you need a two-dimensional coordinate system that tracks both horizontal and vertical positions simultaneously.
How do we express both where something is and where it moved using clean mathematical notation?

The Position Vector

A position vector (written as r⃗\vec{r}) is an arrow drawn from a chosen reference point, called the origin (0,0)(0,0), directly to the location of the object (x,y)(x, y).
We break r⃗\vec{r} into perpendicular components using the standard unit vectors i^\hat{i} (one unit along the +x+x axis) and j^\hat{j} (one unit along the +y+y axis), giving r⃗=xi^+yj^\vec{r} = x\hat{i} + y\hat{j}.
Because the xx and yy axes meet at a right angle, you can find the straight-line distance from the originβ€”the magnitude ∣rβƒ—βˆ£|\vec{r}|β€”using the Pythagorean theorem: ∣rβƒ—βˆ£=x2+y2|\vec{r}| = \sqrt{x^2 + y^2}.
Its direction angle θ\theta measured counterclockwise from the positive xx-axis follows basic trigonometry: tan⁑θ=yx\tan\theta = \frac{y}{x}, so θ=arctan⁑(yx)\theta = \arctan\left(\frac{y}{x}\right).
Position tells you where an object is at one instantβ€”but what happens when that object moves from one point to another along a winding path?

Displacement in Two Dimensions

Displacement (written as Δr⃗\Delta\vec{r}) represents the straight-line change in position from an initial position r⃗i\vec{r}_i to a final position r⃗f\vec{r}_f, completely ignoring the actual path taken.
Mathematically, displacement is final position minus initial position: Ξ”rβƒ—=rβƒ—fβˆ’rβƒ—i=(xfβˆ’xi)i^+(yfβˆ’yi)j^\Delta\vec{r} = \vec{r}_f - \vec{r}_i = (x_f - x_i)\hat{i} + (y_f - y_i)\hat{j}.