VECTOR FUNCTIONS

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14
VECTOR FUNCTIONS
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VECTOR FUNCTIONS

• The functions that we have been using so far
  have been real-valued functions.

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VECTOR FUNCTIONS

• We now study functions whose values are
  vectorsbecause such functions are needed to
  describe curves and surfaces in space.

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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.

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VECTOR FUNCTIONS
14.1 Vector Functions and Space Curves
In this section, we will learn about Vector
functions and drawing their corresponding space
curves.
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FUNCTION

• In general, a function is a rule that assigns to
  each element in the domain an element in the
  range.

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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.

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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).

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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

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VECTOR FUNCTIONS

• We usually use the letter t to denote the
  independent variable because it represents time
  in most applications of vector functions.

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VECTOR FUNCTIONS
Example 1

• Ifthen the component functions are

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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).

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LIMIT OF A VECTOR

• The limit of a vector function r is defined by
  taking the limits of its component functions as
  follows.

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LIMIT OF A VECTOR
Definition 1

• If r(t) f(t), g(t), h(t), then provided
  the limits of the component functions exist.

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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.

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LIMIT OF A VECTOR

• Equivalently, we could have used an e-d
  definition.
• See Exercise 45.

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LIMIT OF A VECTOR

• Limits of vector functions obey the same rules
  as limits of real-valued functions.
• See Exercise 43.

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LIMIT OF A VECTOR
Example 2

• Find , where

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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)

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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.

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CONTINUOUS VECTOR FUNCTIONS

• There is a close connection between continuous
  vector functions and space curves.

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CONTINUOUS VECTOR FUNCTIONS

• Suppose that f, g, and h are continuous
  real-valued functions on an interval I.

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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.

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PARAMETRIC EQUATIONS

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

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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))

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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.

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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).

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SPACE CURVES
Example 3

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

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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.

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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

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PLANE CURVES

• Plane curves can also be represented in vector
  notation.

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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

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SPACE CURVES
Example 4

• Sketch the curve whose vector equation is
  r(t) cos t i sin t j t k

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SPACE CURVES
Example 4

• The parametric equations for this curve are
  x cos t y sin t z t

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SPACE CURVES
Example 4

• Since x2 y2 cos2t sin2t 1, the curve
  must lie on the circular cylinder x2 y2 1

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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

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HELIX
Example 4

• Since z t, the curve spirals upward around the
  cylinder as t increases.
• The curve is called a helix.

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HELICES

• The corkscrew shape of the helix in Example 4 is
  familiar from its occurrence in coiled springs.

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HELICES

• It also occurs in the model of DNA
  (deoxyribonucleic acid, the genetic material of
  living cells).

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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.

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SPACE CURVES

• In Examples 3 and 4, we were given vector
  equations of curves and asked for a geometric
  description or sketch.

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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.

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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).

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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

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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

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SPACE CURVES
Example 5

• The corresponding parametric equations are
• x 1 t y 3 4t z 2 5t
  where 0 t 1

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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.

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SPACE CURVES
Example 6

• This figure shows how the plane and the cylinder
  intersect.

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SPACE CURVES
Example 6

• This figure shows the curve of intersection C,
  which is an ellipse.

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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

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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

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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.

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SPACE CURVES
Example 6

• The arrows indicate the direction in which C is
  traced as the parameter t increases.

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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.

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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

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TOROIDAL SPIRAL

• Its called a toroidal spiral because it lies on
  a torus.

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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

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SPACE CURVES BY COMPUTERS

• It wouldnt be easy to plot either of these
  curves by hand.

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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.

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SPACE CURVES BY COMPUTERS

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

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SPACE CURVES BY COMPUTERS

• The next example shows how to cope with this
  problem.

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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.

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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

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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.

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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.

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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.

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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.

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SPACE CURVES BY COMPUTERS
Example 7

• We get an even better idea of the curve when we
  view it from different vantage points.

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SPACE CURVES BY COMPUTERS
Example 7

• This figure shows the result of rotating the box
  to give another viewpoint.

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SPACE CURVES BY COMPUTERS
Example 7

• These figures show the views we get when we look
  directly at a face of the box.

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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.

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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.

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SPACE CURVES BY COMPUTERS

• Another method of visualizing a space curve is
  to draw it on a surface.

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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.

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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.

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SPACE CURVES BY COMPUTERS

• We also used this method in Example 4 to
  visualize the helix lying on the circular
  cylinder.

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SPACE CURVES BY COMPUTERS

• A third method for visualizing the twisted cubic
  is to realize that it also lies on the cylinder
  z x3.

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SPACE CURVES BY COMPUTERS

• So, it can be viewed as the curve of
  intersection of the cylinders y x2 and z
  x3

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SPACE CURVES BY COMPUTERS

• We have seen that an interesting space curve,
  the helix, occurs in the model of DNA.

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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.

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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.

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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

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SPACE CURVES BY COMPUTERS

• It can be a curve whose projection is the
  trochoid investigated in Exercise 40 in Section
  10.1

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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.

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SPACE CURVES BY COMPUTERS

• For example, the new figure shows the curve of
  the previous figure as rendered by the tubeplot
  command in Maple.

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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/