Vector-Valued Functions

Part 1 of 7 — Position, Velocity, Acceleration

Vector Position

Vectors 🎯

Key Takeaways — Part 1

Differentiate component-wise. Speed = magnitude of velocity.

Vector-Valued Functions

Part 2 of 7 — Integration of Vectors

Integrating Vector Functions

Vector Integration 🎯

Key Takeaways — Part 2

Integrate vectors component-by-component. Don't forget initial conditions!

Vector-Valued Functions

Part 3 of 7 — Distance Traveled

Total Distance

ext{Distance} = int_a^b |ec{v}(t)|,dt = int_a^b sqrt{[x'(t)]^2 + [y'(t)]^2},dt

Displacement vs Distance

Displacement (net change): ec{r}(b) - ec{r}(a)

Distance (total path length): int_a^b |ec{v}(t)|,dt

Distance 🎯

Key Takeaways — Part 3

Distance ≠ displacement. Distance integrates speed; displacement is net change.

Vector-Valued Functions

Part 4 of 7 — Motion Analysis

Direction of Motion

The velocity vector ec{v}(t) points in the direction of motion.

When Is the Particle at Rest?

At rest when ec{v}(t) = ec{0}, meaning both $x'(t) = 0$ and $y'(t) = 0$ simultaneously.

Motion Analysis 🎯

Key Takeaways — Part 4

A particle is at rest only when ALL velocity components are zero simultaneously.

Vector-Valued Functions

Part 5 of 7 — Acceleration & Tangent/Normal

Tangent Vector

hat{T}(t) = rac{ec{v}(t)}{|ec{v}(t)|}

Acceleration decomposition (BC topic)

ec{a} can be decomposed into tangential and normal components:

• Tangential a_T = rac{d}{dt}|ec{v}| — changes speed
• Normal $a_N$ — changes direction

Acceleration 🎯

Key Takeaways — Part 5

For uniform circular motion, acceleration is centripetal (toward center).

Vector-Valued Functions

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Vector-Valued Functions — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Vector-Valued Functions — Complete! ✅