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14

VECTOR FUNCTIONS

VECTOR FUNCTIONS

- The functions that we have been using so far

have been real-valued functions.

VECTOR FUNCTIONS

- We now study functions whose values are

vectorsbecause such functions are needed to

describe curves and surfaces in space.

VECTOR FUNCTIONS

- We will also use vector-valued functions to

describe the motion of objects through space. - In particular, we will use them to derive

Keplers laws of planetary motion.

VECTOR FUNCTIONS

14.1 Vector Functions and Space Curves

In this section, we will learn about Vector

functions and drawing their corresponding space

curves.

FUNCTION

- In general, a function is a rule that assigns to

each element in the domain an element in the

range.

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

- We are most interested in vector functions r

whose values are three-dimensional (3-D) vectors. - This means that, for every number t in the

domain of r, there is a unique vector in V3

denoted by r(t).

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

VECTOR FUNCTIONS

- We usually use the letter t to denote the

independent variable because it represents time

in most applications of vector functions.

VECTOR FUNCTIONS

Example 1

- Ifthen the component functions are

VECTOR FUNCTIONS

Example 1

- By our usual convention, the domain of r consists

of all values of t for which the expression for

r(t) is defined. - The expressions t3, ln(3 t), and are all

defined when 3 t gt 0 and t 0. - Therefore, the domain of r is the interval 0, 3).

LIMIT OF A VECTOR

- The limit of a vector function r is defined by

taking the limits of its component functions as

follows.

LIMIT OF A VECTOR

Definition 1

- If r(t) f(t), g(t), h(t), then provided

the limits of the component functions exist.

LIMIT OF A VECTOR

- If , this definition is

equivalent to saying that the length and

direction of the vector r(t) approach the length

and direction of the vector L.

LIMIT OF A VECTOR

- Equivalently, we could have used an e-d

definition. - See Exercise 45.

LIMIT OF A VECTOR

- Limits of vector functions obey the same rules

as limits of real-valued functions. - See Exercise 43.

LIMIT OF A VECTOR

Example 2

- Find , where

LIMIT OF A VECTOR

Example 2

- According to Definition 1, the limit of r is the

vector whose components are the limits of the

component functions of r - (Equation 2 in Section 3.3)

CONTINUOUS VECTOR FUNCTION

- 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.

CONTINUOUS VECTOR FUNCTIONS

- There is a close connection between continuous

vector functions and space curves.

CONTINUOUS VECTOR FUNCTIONS

- Suppose that f, g, and h are continuous

real-valued functions on an interval I.

SPACE CURVE

Equations 2

- 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.

PARAMETRIC EQUATIONS

- Equations 2 are called parametric equations of C.
- Also, t is called a parameter.

SPACE CURVES

- We can think of C as being traced out by a

moving particle whose position at time t is

(f(t), g(t), h(t))

SPACE CURVES

- If we now consider the vector function r(t)

f(t), g(t), h(t), then r(t) is the position

vector of the point P(f(t), g(t), h(t)) on C.

SPACE CURVES

- Thus, any continuous vector function r defines a

space curve C that is traced out by the tip of

the moving vector r(t).

SPACE CURVES

Example 3

- Describe the curve defined by the vector function
- r(t) 1 t, 2 5t, 1 6t

SPACE CURVES

Example 3

- The corresponding parametric equations are x

1 t y 2 5t z 1 6t - We recognize these from Equations 2 of Section

12.5 as parametric equations of a line passing

through the point (1, 2 , 1) and parallel to

the vector 1, 5, 6.

SPACE CURVES

Example 3

- Alternatively, we could observe that the

function can be written as r r0 tv, where r0

1, 2 , 1 and v 1, 5, 6. - This is the vector equation of a line as given

by Equation 1 of Section 12.5

PLANE CURVES

- Plane curves can also be represented in vector

notation.

PLANE CURVES

- For instance, the curve given by the parametric

equations x t2 2t and y t 1

could also be described by the vector equation

r(t) t2 2t, t 1 (t2 2t) i

(t 1) j where i 1, 0 and j 0, 1

SPACE CURVES

Example 4

- Sketch the curve whose vector equation is

r(t) cos t i sin t j t k

SPACE CURVES

Example 4

- The parametric equations for this curve are

x cos t y sin t z t

SPACE CURVES

Example 4

- Since x2 y2 cos2t sin2t 1, the curve

must lie on the circular cylinder x2 y2 1

SPACE CURVES

Example 4

- The point (x, y, z) lies directly above the

point (x, y, 0). - This other point moves counterclockwise around

the circle x2 y2 1 in the xy-plane. - See Example 2 in Section 10.1

HELIX

Example 4

- Since z t, the curve spirals upward around the

cylinder as t increases. - The curve is called a helix.

HELICES

- The corkscrew shape of the helix in Example 4 is

familiar from its occurrence in coiled springs.

HELICES

- It also occurs in the model of DNA

(deoxyribonucleic acid, the genetic material of

living cells).

HELICES

- In 1953, James Watson and Francis Crick showed

that the structure of the DNA molecule is that of

two linked, parallel helixes that are

intertwined.

SPACE CURVES

- In Examples 3 and 4, we were given vector

equations of curves and asked for a geometric

description or sketch.

SPACE CURVES

- In the next two examples, we are given a

geometric description of a curve and are asked to

find parametric equations for the curve.

SPACE CURVES

Example 5

- Find a vector equation and parametric equations

for the line segment that joins the point P(1,

3, 2) to the point Q(2, 1, 3).

SPACE CURVES

Example 5

- In Section 12.5, we found a vector equation for

the line segment that joins the tip of the

vector r0 to the tip of the vector r1 - r(t) (1 t) r0 t r1 0 t 1
- See Equation 4 of Section 12.5

SPACE CURVES

Example 5

- Here, we take r0 1, 3 , 2 and r1 2 ,

1, 3 to obtain a vector equation of the line

segment from P to Q - or

SPACE CURVES

Example 5

- The corresponding parametric equations are
- x 1 t y 3 4t z 2 5t

where 0 t 1

SPACE CURVES

Example 6

- Find a vector function that represents the curve

of intersection of the cylinder x2 y2 1 and

the plane y z 2.

SPACE CURVES

Example 6

- This figure shows how the plane and the cylinder

intersect.

SPACE CURVES

Example 6

- This figure shows the curve of intersection C,

which is an ellipse.

SPACE CURVES

Example 6

- The projection of C onto the xy-plane is the

circle x2 y2 1, z 0. - So, we know from Example 2 in Section 10.1 that

we can write x cos t y sin t where 0

t 2p

SPACE CURVES

Example 6

- From the equation of the plane, we have
- z 2 y 2 sin t
- So, we can write parametric equations for C as

x cos t y sin t z 2 sin twhere 0 t

2p

PARAMETRIZATION

Example 6

- The corresponding vector equation is r(t)

cos t i sin t j (2 sin t) k where 0

t 2p - This equation is called a parametrization of the

curve C.

SPACE CURVES

Example 6

- The arrows indicate the direction in which C is

traced as the parameter t increases.

USING COMPUTERS TO DRAW SPACE CURVES

- Space curves are inherently more difficult to

draw by hand than plane curves. - For an accurate representation, we need to use

technology.

USING COMPUTERS TO DRAW SPACE CURVES

- This figure shows a computer-generated graph of

the curve with the following parametric

equations - x (4 sin 20t) cos t
- y (4 sin 20t) sin t
- z cos 20 t

TOROIDAL SPIRAL

- Its called a toroidal spiral because it lies on

a torus.

TREFOIL KNOT

- Another interesting curve, the trefoil knot, is

graphed here. - It has the equations
- x (2 cos 1.5 t) cos t
- y (2 cos 1.5 t) sin t
- z sin 1.5 t

SPACE CURVES BY COMPUTERS

- It wouldnt be easy to plot either of these

curves by hand.

SPACE CURVES BY COMPUTERS

- Even when a computer is used to draw a space

curve, optical illusions make it difficult to get

a good impression of what the curve really looks

like.

SPACE CURVES BY COMPUTERS

- This is especially true in this figure.
- See Exercise 44.

SPACE CURVES BY COMPUTERS

- The next example shows how to cope with this

problem.

TWISTED CUBIC

Example 7

- Use a computer to draw the curve with vector

equation r(t) t, t2,

t3 - This curve is called a twisted cubic.

SPACE CURVES BY COMPUTERS

Example 7

- We start by using the computer to plot the curve

with parametric equations x t, y t2, z

t3 for -2 t 2

SPACE CURVES BY COMPUTERS

Example 7

- The result is shown here.
- However, its hard to see the true nature of the

curve from this graph alone.

SPACE CURVES BY COMPUTERS

Example 7

- Most 3-D computer graphing programs allow the

user to enclose a curve or surface in a box

instead of displaying the coordinate axes.

SPACE CURVES BY COMPUTERS

Example 7

- When we look at the same curve in a box, we

have a much clearer picture of the curve.

SPACE CURVES BY COMPUTERS

Example 7

- We can see that
- It climbs from a lower corner of the box to the

upper corner nearest us. - It twists as it climbs.

SPACE CURVES BY COMPUTERS

Example 7

- We get an even better idea of the curve when we

view it from different vantage points.

SPACE CURVES BY COMPUTERS

Example 7

- This figure shows the result of rotating the box

to give another viewpoint.

SPACE CURVES BY COMPUTERS

Example 7

- These figures show the views we get when we look

directly at a face of the box.

SPACE CURVES BY COMPUTERS

Example 7

- In particular, this figure shows the view from

directly above the box. - It is the projection of the curve on the

xy-plane, namely, the parabola y x2.

SPACE CURVES BY COMPUTERS

Example 7

- This figure shows the projection on the

xz-plane, the cubic curve z x3. - Its now obvious why the given curve is called

a twisted cubic.

SPACE CURVES BY COMPUTERS

- Another method of visualizing a space curve is

to draw it on a surface.

SPACE CURVES BY COMPUTERS

- For instance, the twisted cubic in Example 7

lies on the parabolic cylinder y x2. - Eliminate the parameter from the first two

parametric equations, x t and y t2.

SPACE CURVES BY COMPUTERS

- This figure shows both the cylinder and the

twisted cubic. - We see that the curve moves upward from the

origin along the surface of the cylinder.

SPACE CURVES BY COMPUTERS

- We also used this method in Example 4 to

visualize the helix lying on the circular

cylinder.

SPACE CURVES BY COMPUTERS

- A third method for visualizing the twisted cubic

is to realize that it also lies on the cylinder

z x3.

SPACE CURVES BY COMPUTERS

- So, it can be viewed as the curve of

intersection of the cylinders y x2 and z

x3

SPACE CURVES BY COMPUTERS

- We have seen that an interesting space curve,

the helix, occurs in the model of DNA.

SPACE CURVES BY COMPUTERS

- Another notable example of a space curve in

science is the trajectory of a positively charged

particle in orthogonally oriented electric and

magnetic fields E and B.

SPACE CURVES BY COMPUTERS

- Depending on the initial velocity given the

particle at the origin, the path of the particle

is either of two curves, as follows.

SPACE CURVES BY COMPUTERS

- It can be a space curve whose projection on the

horizontal plane is the cycloid we studied in

Section 10.1

SPACE CURVES BY COMPUTERS

- It can be a curve whose projection is the

trochoid investigated in Exercise 40 in Section

10.1

SPACE CURVES BY COMPUTERS

- Some computer algebra systems provide us with a

clearer picture of a space curve by enclosing it

in a tube. - Such a plot enables us to see whether one part

of a curve passes in front of or behind another

part of the curve.

SPACE CURVES BY COMPUTERS

- For example, the new figure shows the curve of

the previous figure as rendered by the tubeplot

command in Maple.

SPACE CURVES BY COMPUTERS

- For further details concerning the physics

involved and animations of the trajectories of

the particles, see the following websites - www.phy.ntnu.edu.tw/java/emField/emField.html
- www.physics.ucla.edu/plasma-exp/Beam/

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