CALC RESEMBLING MULTIVARIABLE
(2D MOTION) EXTENTION
Before we soar into three dimensions, let’s revisit 2D
parametric motion—the foundation for everything that follows.
INTRODUCTION
In our study of 3D motion, we explore how objects
move through space with respect to time,
analyzing their position, velocity, and acceleration
in three dimensions. The graph above illustrates
the flight paths of a projectile in both controlled
and uncontrolled conditions, showing how real
world objects, such as missiles, drones, or space- Notations For 2D
bound rockets, follow complex spatial trajectories.
Component Form
This visualization involves motion along three
Unit Vector Form
axes: range (x), cross-range (y), and height (z),
forming a full 3D path. Using parametric equations
such as r(t)=<x(t), y(t), z(t)>, we can model these
motion and compute key features like speed, arc
length, and curvature. This forms the foundation of
vector calculus in physics and engineering, where
understanding how an object moves not just
forward but also up and sideways is critical.
For Question 2:
APPLICATION 1. Formula for curvature:
v=
MULTIVARIABLE CALCULUS
3D MOTION IN
PROBLEM
3î
+4
2. Differentiate r(t) to sollve velocity and acceleration:
ĵ+
t 2 k̂
A shuttlecock is hit upward
with spin, causing it to
3. Compute magnitude of velocity:
curve in space. Its motion
is modeled by the 3D
position function:
4. Compute cross product:
Q1: Assume the net lies in the plane x=2. Find the
minimum vertical distance from the shuttlecock to
the top of the net (z=1.5) as it crosses the net.
5. Final Curvature Formula:
Q2: What is the curvature κ(t) of the path?
CONCLUSION
Just as suggested by the application
problem, 3D motion has many real-world
SOLUTION
applications across science, engineering,
and daily life. These applications rely on
multivariable calculus to accurately describe
and predict movement through space. By
For Question 1: mastering 3D parametric motion, we gain
the tools to model everything from planetary
orbits to AI-driven robotics.
References
https://physicsteacher.in/2021/07/22/motion-in-2-and-3-dimensions-formula-
Short Pamphlet by
numerical
https://www.khanacademy.org/math/multivariable-calculus
https://www.researchgate.net/publication/317122654_Two_real-
world_case_studies_on_3D_web_applications_for_participatory_urban_pla Max, Lanson, Alex, and Ricky
nning
https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/
