Vector-Valued Functions
Tom Green
6 min read
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Study Guide Overview
This study guide covers vector-valued functions and their application to planar motion. It explains position vectors p(t), including magnitude and notation, and velocity vectors v(t), including component interpretation (horizontal and vertical velocity) and magnitude (speed). It emphasizes the relationship between position and velocity as a key concept and provides practice questions on finding these vectors, determining direction of motion, and calculating speed.
#Vector-Valued Functions: Your Guide to Planar Motion 🚀
Hey there! Let's dive into vector-valued functions, where we combine parametric functions, planar motion, and vectors to describe how things move in 2D. Think of it as unlocking the secrets of motion! 🔑
#Position Vector
#What is it?
The position vector, denoted as p(t), tells you exactly where a particle is at any given time, t. It's like a GPS for a moving object! We can express it in two ways:
- p(t) = x(t)i + y(t)j: Here, x(t) and y(t) are the coordinates of the particle at time t, and i and j are the unit vectors in the x and y directions. Think of it as breaking down the position into its horizontal and vertical components.
- p(t) = < x(t), y(t) >: This is just another way to write the same thing, using vector notation. It's often more compact and easier to work with.
The magnitude of the position vector, |p(t)|, gives the distance of the particle from the origin (0,0) at time t. It's like measuring how far the particle is from the starting point. 📏
#Velocity Vector
#What is it?
The velocity vector, denoted as v(t), describes how fast and in what direction a particle is moving at any time t. It's like the speedometer and compass of the particle! 🧭
- v(t) = < x'(t), y'(t) >: The x...
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