Key Idea: Vectors describe things that move in straight lines, like birds or planes. You find where they are, how fast they go, whether two of them meet, and how close they get.
| Quantity | Rule |
|---|---|
| Position | r(t) = r₀ + t·v |
| Velocity | The vector multiplying t |
| Speed | |v| = √(vₓ² + vy²), one positive number |
Collision or closest approach
- Collision — Set the two position vectors equal. One t must fit every coordinate.
- Distance — Form d(t) = |rB(t) − rA(t)|.
- Closest — Graph d(t) on the GDC. The minimum gives the time and the least distance.
- Interpret — Say whether a safety rule, such as 5 km, is met.
To meet, two objects need the same place at the same time.
A bird has r(t) = (−5 + 8t, 10 + 6t). What is its speed? |(8, 6)| = 10 km/h.
Where is the bird after 4 hours? (27, 34).
Two paths cross. Must the objects collide? No. They may reach the crossing at different times.
Two planes come within 4.47 km. The rule is 5 km. Is it met? No. They come too close.