1. 13
VECTOR FUNCTIONS

2. VECTOR FUNCTIONS
The functions that we have been using so far have been real-valued functions.

3. VECTOR FUNCTIONS
We now study functions whose values are vectors—because such functions are needed to describe curves and surfaces in space.

4. 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 Kepler’s laws of planetary motion.

5. Vector Functions and Space Curves
13.1
Vector Functions and Space Curves
In this section, we will learn about:
Vector functions and drawing
their corresponding space curves.

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

7. 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.

8. 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).

9. We can write: r(t) = ‹f(t), g(t), h(t)› = f(t) i + g(t) j + h(t) k
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

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

11. If then the component functions are:
VECTOR FUNCTIONS
Example 1
If then the component functions are:

12. 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 > 0 and t ≥ 0.
Therefore, the domain of r is the interval [0, 3).

13. LIMIT OF A VECTOR
The limit of a vector function r is defined by taking the limits of its component functions as follows.

14. LIMIT OF A VECTOR
Definition 1
If r(t) = ‹f(t), g(t), h(t)›, then provided the limits of the component functions exist.

15. 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.

16. Equivalently, we could have used an ε-δ definition.
LIMIT OF A VECTOR
Equivalently, we could have used an ε-δ definition.
See Exercise 45.

17. LIMIT OF A VECTOR
Limits of vector functions obey the same rules as limits of real-valued functions.
See Exercise 43.

18. LIMIT OF A VECTOR
Example 2
Find , where

19. 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)

20. 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.

21. CONTINUOUS VECTOR FUNCTIONS
There is a close connection between continuous vector functions and space curves.

22. CONTINUOUS VECTOR FUNCTIONS
Suppose that f, g, and h are continuous real-valued functions on an interval I.

23. 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.

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

25. 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))

26. 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.

27. 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).

28. Describe the curve defined by the vector function
SPACE CURVES
Example 3
Describe the curve defined by the vector function
r(t) = ‹1 + t, 2 + 5t, –1 + 6t›

29. 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 as parametric equations of a line passing through the point (1, 2 , –1) and parallel to the vector ‹1, 5, 6›.

30. 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

31. Plane curves can also be represented in vector notation.

32. 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›

33. SPACE CURVES
Example 4
Sketch the curve whose vector equation is: r(t) = cos t i + sin t j + t k

34. The parametric equations for this curve are: x = cos t y = sin t z = t
SPACE CURVES
Example 4
The parametric equations for this curve are: x = cos t y = sin t z = t

35. SPACE CURVES
Example 4
Since x2 + y2 = cos2t + sin2t = 1, the curve must lie on the circular cylinder x2 + y2 = 1

36. The point (x, y, z) lies directly above the point (x, y, 0).
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

37. HELIX
Example 4
Since z = t, the curve spirals upward around the cylinder as t increases.
The curve is called a helix.

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

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

40. 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.

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

42. 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.

43. 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).

44. 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

45. 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

46. The corresponding parametric equations are:
SPACE CURVES
Example 5
The corresponding parametric equations are:
x = 1 + t y = 3 – 4t z = – 2 + 5t where 0 ≤ t ≤ 1

47. 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.

48. This figure shows how the plane and the cylinder intersect.
SPACE CURVES
Example 6
This figure shows how the plane and the cylinder intersect.

49. This figure shows the curve of intersection C, which is an ellipse.
SPACE CURVES
Example 6
This figure shows the curve of intersection C, which is an ellipse.

50. 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 that we can write: x = cos t y = sin t where 0 ≤ t ≤ 2π

51. From the equation of the plane, we have:
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 t where 0 ≤ t ≤ 2π

52. PARAMETRIZATION
Example 6
The corresponding vector equation is: r(t) = cos t i + sin t j + (2 – sin t) k where 0 ≤ t ≤ 2π
This equation is called a parametrization of the curve C.

53. SPACE CURVES
Example 6
The arrows indicate the direction in which C is traced as the parameter t increases.

54. 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.

55. 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

56. It’s called a toroidal spiral because it lies on a torus.

57. 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

58. SPACE CURVES BY COMPUTERS
It wouldn’t be easy to plot either of these curves by hand.

59. 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.

60. SPACE CURVES BY COMPUTERS
This is especially true in this figure.
See Exercise 44.

61. SPACE CURVES BY COMPUTERS
The next example shows how to cope with this problem.

62. 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.

63. 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

64. SPACE CURVES BY COMPUTERS
Example 7
The result is shown here.
However, it’s hard to see the true nature of the curve from this graph alone.

65. 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.

66. 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.

67. 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.

68. SPACE CURVES BY COMPUTERS
Example 7
We get an even better idea of the curve when we view it from different vantage points.

69. SPACE CURVES BY COMPUTERS
Example 7
This figure shows the result of rotating the box to give another viewpoint.

70. SPACE CURVES BY COMPUTERS
Example 7
These figures show the views we get when we look directly at a face of the box.

71. 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.

72. SPACE CURVES BY COMPUTERS
Example 7
This figure shows the projection on the xz-plane, the cubic curve z = x3.
It’s now obvious why the given curve is called a twisted cubic.

73. SPACE CURVES BY COMPUTERS
Another method of visualizing a space curve is to draw it on a surface.

74. 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.

75. 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.

76. SPACE CURVES BY COMPUTERS
We also used this method in Example 4 to visualize the helix lying on the circular cylinder.

77. SPACE CURVES BY COMPUTERS
A third method for visualizing the twisted cubic is to realize that it also lies on the cylinder z = x3.

78. SPACE CURVES BY COMPUTERS
So, it can be viewed as the curve of intersection of the cylinders y = x2 and z = x3

79. SPACE CURVES BY COMPUTERS
We have seen that an interesting space curve, the helix, occurs in the model of DNA.

80. 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.

81. 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.

82. 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

83. SPACE CURVES BY COMPUTERS
It can be a curve whose projection is the trochoid investigated in Exercise 40 in Section 10.1

84. 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.

85. SPACE CURVES BY COMPUTERS
For example, the new figure shows the curve of the previous figure as rendered by the tubeplot command in Maple.

86. SPACE CURVES BY COMPUTERS
For further details concerning the physics involved and animations of the trajectories of the particles, see the following websites: