Chapter 13 – Vector Functions 13.1 Vector Functions and Space Curves 1 Objectives:  Use vector -valued functions to describe the motion of objects through space  Draw vector functions and their corresponding space curves © jdannels -

Vector Functions The functions that we have been using so far have been real-valued functions. We study functions whose values are vectors because such functions are needed to describe curves and surfaces in space. We will also use vector-valued functions to describe the motion of objects through space. ◦ In particular, we will use them to derive Kepler’s laws of planetary motion Vector Functions and Space Curves2

Definition – Vector function A vector-valued function, or vector function, is simply a function whose: ◦ Domain is a set of real numbers. ◦ Range is a set of vectors Vector Functions and Space Curves3

Example 1 – pg. 845 #2 Find the domain of the vector function Vector Functions and Space Curves4

Definition – Component Functions If f(t), g(t), and h(t) are the components of the vector r(t), then f, g, and h are real-valued functions called the component functions of r. We can write: r(t) = ‹ f(t), g(t), h(t) › = f(t) i + g(t) j + h(t) k Note: We usually use the letter t to denote the independent variable because it represents time in most applications of vector functions Vector Functions and Space Curves5

Definition – Limit of a Vector The limit of a vector function r is defined by taking the limits of its component functions as follows Vector Functions and Space Curves6

Definition - Continuous A vector function r is continuous at a if: ◦ In view of Definition 1, we see that r is continuous at a if and only if its component functions f, g, and h are continuous at a Vector Functions and Space Curves7

Example 2 Find the limit 13.1 Vector Functions and Space Curves8

Space Curve Then, the set C of all points (x, y,z) in space, where x = f(t) y = g(t) z = h(t) and t varies throughout the interval I is called a space curve Vector Functions and Space Curves9 The equations are called parametric equations of C. t is called a parameter.

Visualization Vector Functions and Space Curves The Twisted Cubic Curve Visualizing Space Curves 13.1 Vector Functions and Space Curves10

Example 3 Find a vector equation and parametric equations for the line segment that joins P to Q. P (-2, 4, 0) Q (6, -1, 2) 13.1 Vector Functions and Space Curves11

Example 4 – pg. 846 # 29 At what points does the curve intersect the paraboloid ? 13.1 Vector Functions and Space Curves12

Example 5 – pg 847 # 47 If two objects travel through space along two different curves, it’s often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of the two particles are given by the vector functions for t  0. Do the particles collide? 13.1 Vector Functions and Space Curves13

More Examples The video examples below are from section 13.1 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 3 Example 3 ◦ Example 6 Example Vector Functions and Space Curves14

Demonstrations Feel free to explore these demonstrations below. Four Space Curves Equation of a Line in Vector Form 2D 13.1 Vector Functions and Space Curves15