Ch 13.1 - Vector functions and space curves

In this section, we will

I define vector valued functions

I look at space curves of certain vector functions

DefinitionA vector-valued function, or vector function, is a function whosedomain is a set of real numbers and whose range is a set of vectors.If f (t), g(t), and h(t) are the components of the vector r(t), thenf , g , and h are real-valued functions called the componentfunctions of r and we can write

r(t) =< f (t), g(t), h(t) >= f (t)i + g(t)j + h(t)k

Example) Let r(t) =< t, t, t >= ti + tj + tk. The componentfunctions are

The domain of r consists of all values of t for which the expressionfor r(t) is defined. For this example, all the component functionsare define for all t ∈ R.

Therefore, the domain of r(t) is

ExampleFind the domain of the vector function

r(t) =

(t − 2

t + 2

)i + sin t j + ln(16− t2)k

The limit of r(t)The limit of a vector function r(t) is defined by taking the limits ofits component functions as follows.If r(t) =< f (t), g(t), h(t) >, then

limt→a

r(t) =< limt→a

f (t), limt→a

g(t), limt→a

h(t) >

Limits of vector functions obey the same rules as limits ofreal-valued functions.

Note: r(t) is continuous at t = a if limt→a r(t) = r(a).

Example) Find the limit and determine whether the continuity at a.

1. limt→0 〈t, sin t, cos t〉

2. limt→0

(e−t i + 3j + sin t

t k)

Space curveSuppose that f , g , and h are continuous real-valued functions onan interval I .

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.

The above equations are called the parametric equations of C andt is a parameter.

Think of C as being traced out by a moving particle whoseposition at time t is (f (t), g(t), h(t)).

ExampleFind a vector equation and parametric equations for the linesegment that joins the point P = (1, 2, 3) to the pointQ = (2, 1− 3).

Note: A vector equation for the line segment that joins the tip ofthe vector r0 to the tip of the vector r1 is:

r(t) = (1− t)r0 + tr1 0 ≤ t ≤ 1

ExampleDescribe the curve defined by the vector function

r(t) = (1 + t)i + (2− t)j + (−1 + 2t)k

Hint: The parametric equations for this curve should look familiar.

ExampleSketch the curve whose vector equation is

r(t) = cos ti + sin tj + tk

Note: The parametric equations for this curve C are

x = cos t, y = sin t, z = t

Solution - Helix

ExampleSketch the curve with the given vector equation. Indicate with anarrow the direction in which t increases.

r(t) =< sin 6t, t >