2.2 Vector-Valued Functions of One Variable

Vector-Valued Functions of One Variable

Introduction

Vector-Valued Functions

Parameterizing Curves

After graduating, you decide to start a self-driving car company because it didn't look too hard based on a few articles you scrolled past online. You decide to call your company Steer Clear and start working on the self-driving part of the car. Your first task is to understand how the location tracking equipment you bought online works. There are two parts to your location tracking system: a receiver box and a tracker that you put in the object you want to track.

After putting the tracker in your pocket, you get in a car and drive around the receiver box. When you download the data about your drive, you notice that the software is outputting a vector from the receiver box to the tracker. The software also seems to love math, because these vectors are written in terms of trig functions.

Below are some of the output vectors from the software. Evaluate each of these vectors and draw these vectors with initial point at the origin on the axes in . -$\langle \cos(0), \sin(0) \rangle$-$\langle \cos\left(\frac{\pi}{2}\right), \sin\left(\frac{\pi}{2}\right)\rangle$-$\langle \cos(\pi), \sin(\pi) \rangle$-$\langle \cos\left(\frac{3\pi}{2}\right), \sin\left(\frac{3\pi}{2}\right)\rangle$

$\vr(t) = \langle \sin(t), \cos(t) \rangle$

$\vr(t) = \langle \sin(2t), \cos(2t) \rangle$

$\vr(t) = \langle \cos(t+\pi), \sin(t+\pi) \rangle$

$\vr(t) = \langle \cos(t^2), \sin(t^2) \rangle$

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

$\vr(t) = \langle t\cos(t), t\sin(t) \rangle$

$\vr(t) = \langle \sin(t)\cos(t), t\sin(t) \rangle$

$\vr(t) = \langle \sin(5t), \sin(4t) \rangle$

$\vr(t) = \langle t^2\sin(t)\cos(t), 0.9t\cos(t^2), \sin(t) \rangle$. (Note that this defines a curve in three-dimensional space.)

Work through this exercise and explain your reasoning step by step.

Parameterize the intersection of the paraboloid with the fundamental plane $x=2$. What type of curve is the intersection?

Parameterize the intersection of the paraboloid with the fundamental plane $y = -1$. What type of curve is the intersection?

Parameterize the intersection of the paraboloid with the fundamental plane $z = 25$. What type of curve is the intersection?

How do your responses change to all three of the preceding questions if you instead consider the surface defined by $z = x^2 - y^2$?