1. Vector-valued Functions

2. Vector-valued Functions
A parametric curve in R3 is a function C from the
real numbers to R3, i.e C: R  R3.

3. Vector-valued Functions
A parametric curve in R3 is a function C from the
real numbers to R3, i.e C: R  R3. It may be
written as:
C(t) = (x(t), y(t), z(t)).
C(t)=(x(t),y(t),z(t))
y
x

4. Vector-valued Functions
A parametric curve in R3 is a function C from the
real numbers to R3, i.e C: R  R3. It may be
written as:
C(t) = (x(t), y(t), z(t)).
The functions x(t), y(t), z(t) are called the
parametric equations or the coordinate functions.
C(t)=(x(t),y(t),z(t))
y
x

5. Vector-valued Functions
A parametric curve in R3 is a function C from the
real numbers to R3, i.e C: R  R3. It may be
written as:
C(t) = (x(t), y(t), z(t)).
The functions x(t), y(t), z(t) are called the
parametric equations or the coordinate functions.
C(t)=(x(t),y(t),z(t))
y
x
C(t) is said to be continuous if x(t), y(t), z(t) are
continuous.

6. Vector-valued Functions
We may view a point (x(t), y(t), z(t)) as the vector
<x(t), y(t), z(t)>.

7. Vector-valued Functions
We may view a point (x(t), y(t), z(t)) as the vector
<x(t), y(t), z(t)>. Using vector notation, we write
C(t) =<x(t), y(t), z(t)> = x(t)i + y(t)j + z(t)k.
This is called a vector-valued function.
C(t) = x(t)i + y(t)j + z(t)k
y
x

8. Vector-valued Functions
We may view a point (x(t), y(t), z(t)) as the vector
<x(t), y(t), z(t)>. Using vector notation, we write
C(t) =<x(t), y(t), z(t)> = x(t)i + y(t)j + z(t)k.
This is called a vector-valued function.
The difference between
parametric form and
vectored-valued form is that C(t) = x(t)i + y(t)j + z(t)k
we may utilize the geometry y
and algebraic operations x
associated with vectors to
investigate the curve.

9. Vector-valued Functions
We may view a point (x(t), y(t), z(t)) as the vector
<x(t), y(t), z(t)>. Using vector notation, we write
C(t) =<x(t), y(t), z(t)> = x(t)i + y(t)j + z(t)k.
This is called a vector-valued function.
The difference between
parametric form and
vectored-valued form is that C(t) = x(t)i + y(t)j + z(t)k
we may utilize the geometry y
and algebraic operations x
associated with vectors to
investigate the curve.
The norm of C(t) is |C(t)| = √ x2(t) + y2(t) + z2(t).

10. Vector-valued Functions
We may view a point (x(t), y(t), z(t)) as the vector
<x(t), y(t), z(t)>. Using vector notation, we write
C(t) =<x(t), y(t), z(t)> = x(t)i + y(t)j + z(t)k.
This is called a vector-valued function.
The difference between
parametric form and
vectored-valued form is that C(t) = x(t)i + y(t)j + z(t)k
we may utilize the geometry y
and algebraic operations x
associated with vectors to
investigate the curve.
The norm of C(t) is |C(t)| = √ x2(t) + y2(t) + z2(t).
|C(t)| is a real valued function, i.e. |C(t)|: R  R.

11. Vector-valued Functions
Example A. Sketch the curve
C(t) = (2t, 3t, 4t)

12. Vector-valued Functions
Example A. Sketch the curve
C(t) = (2t, 3t, 4t)
This is the line going through
the origin in the direction of
<2, 3, 4>

13. Vector-valued Functions
Example A. Sketch the curve
C(t) = (2t, 3t, 4t) <2, 3, 4>
This is the line going through y
the origin in the direction of x
<2, 3, 4>

14. Vector-valued Functions
Example A. Sketch the curve
C(t) = (2t, 3t, 4t) <2, 3, 4>
This is the line going through y
the origin in the direction of x
<2, 3, 4>
Example B. Sketch the curve
C(t) = (2t+1, 3t+4, 4t+5)

15. Vector-valued Functions
Example A. Sketch the curve
C(t) = (2t, 3t, 4t) <2, 3, 4>
This is the line going through y
the origin in the direction of x
<2, 3, 4>
Example B. Sketch the curve
C(t) = (2t+1, 3t+4, 4t+5)
This is the line going through
the point (1, 4, 5) in the
direction of <2, 3, 4>

16. Vector-valued Functions
Example A. Sketch the curve
C(t) = (2t, 3t, 4t) <2, 3, 4>
This is the line going through y
the origin in the direction of x
<2, 3, 4>
Example B. Sketch the curve
C(t) = (2t+1, 3t+4, 4t+5) (1, 4, 5)
This is the line going through
the point (1, 4, 5) in the y
direction of <2, 3, 4> x

17. Vector-valued Functions
Example C. Sketch the curve
C(t) = (cos(t), 2, sin(t))

18. Vector-valued Functions
Example C. Sketch the curve
C(t) = (cos(t), 2, sin(t))
x(t) = cos(t) and z(t) = sin(t)
form a circle in the xz-plane,
in this case, in the plane y=2.

19. Vector-valued Functions
Example C. Sketch the curve t=π/2:
(0, 2, 1)
C(t) = (cos(t), 2, sin(t))
x(t) = cos(t) and z(t) = sin(t) y
form a circle in the xz-plane, t=0:
in this case, in the plane y=2. x (1, 2, 0)
y=2

20. Vector-valued Functions
Example C. Sketch the curve t=π/2:
(0, 2, 1)
C(t) = (cos(t), 2, sin(t))
x(t) = cos(t) and z(t) = sin(t) y
form a circle in the xz-plane, t=0:
in this case, in the plane y=2. x (1, 2, 0)
y=2
Example D. Sketch the curve
C(t) = (cos(t), t, sin(t))

21. Vector-valued Functions
Example C. Sketch the curve t=π/2:
(0, 2, 1)
C(t) = (cos(t), 2, sin(t))
x(t) = cos(t) and z(t) = sin(t) y
form a circle in the xz-plane, t=0:
in this case, in the plane y=2. x (1, 2, 0)
y=2
Example D. Sketch the curve
C(t) = (cos(t), t, sin(t))
The x(t), z(t) still go around
in circles as t changes.

22. Vector-valued Functions
Example C. Sketch the curve t=π/2:
(0, 2, 1)
C(t) = (cos(t), 2, sin(t))
x(t) = cos(t) and z(t) = sin(t) y
form a circle in the xz-plane, t=0:
in this case, in the plane y=2. x (1, 2, 0)
y=2
Example D. Sketch the curve
C(t) = (cos(t), t, sin(t))
The x(t), z(t) still go around
in circles as t changes.
But the y coordinate pulls
the circle along the y-axis as
t changes.

23. Vector-valued Functions
Example C. Sketch the curve t=π/2:
(0, 2, 1)
C(t) = (cos(t), 2, sin(t))
x(t) = cos(t) and z(t) = sin(t) y
form a circle in the xz-plane, t=0:
in this case, in the plane y=2. x (1, 2, 0)
y=2
Example D. Sketch the curve
C(t) = (cos(t), t, sin(t))
The x(t), z(t) still go around
in circles as t changes.
But the y coordinate pulls
the circle along the y-axis as
t changes. We get a circular coil as the graph.

24. Vector-valued Functions
Example C. Sketch the curve t=π/2:
(0, 2, 1)
C(t) = (cos(t), 2, sin(t))
x(t) = cos(t) and z(t) = sin(t) y
form a circle in the xz-plane, t=0:
in this case, in the plane y=2. x (1, 2, 0)
y=2
Example D. Sketch the curve
C(t) = (cos(t), t, sin(t))
The x(t), z(t) still go around
y
in circles as t changes.
But the y coordinate pulls x
the circle along the y-axis as
t changes. We get a circular coil as the graph.

25. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta

26. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define:
C'(t) = lim 1 [C(t+h) – C(t)]
h0 h
if the limit exists.

27. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x

28. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x

29. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
[C(t+h) – C(t)]
as h0, = C'(t)
h

30. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
[C(t+h) – C(t)]
as h0, = C'(t)
h
y
[C(t+h) – C(t)]
C(t)
x

31. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
[C(t+h) – C(t)]
as h0, = C'(t)
h
y
[C(t+h) – C(t)]
C(t)
x

32. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
* C'(t) is a vector tangent to
and pointing in the same as h0,
[C(t+h) – C(t)]
= C'(t)
h
direction as the curve C(t).
y
[C(t+h) – C(t)]
C(t)
x

33. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
[C(t+h) – C(t)]
as h0, = C'(t)
h
y
C'(t)
[C(t+h) – C(t)]
C(t)
x

34. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
* C'(t) is a vector tangent to
and pointing in the same as h0,
[C(t+h) – C(t)]
= C'(t)
h
direction as the curve C(t).
y
C'(t)
[C(t+h) – C(t)]
C(t)
x

35. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
* C'(t) is a vector tangent to
and pointing in the same as h0,
[C(t+h) – C(t)]
= C'(t)
h
direction as the curve C(t).
y
* The norm |C'(t)| gives the C'(t)
speed of the point traversing C. [C(t+h) – C(t)]
C(t)
x

36. Vector-valued Functions
* We define:
lim C(t) = lim x(t)i + lim y(t)j + lim z(t)k
ta ta ta ta
* We define: y
C'(t) = lim 1 [C(t+h) – C(t)] [C(t+h) – C(t)]
h0 h C(t+h)
if the limit exists. C(t)
x
* C'(t) is a vector tangent to
and pointing in the same as h0,
[C(t+h) – C(t)]
= C'(t)
h
direction as the curve C(t).
y
* The norm |C'(t)| gives the C'(t)
speed of the point traversing C. [C(t+h) – C(t)]
* If C'(t) = 0, we call C'(t) the C(t)
x
tangent vector at C(t).

37. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
C'(t)
C(t)
x

38. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x

39. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x
x(t) = t2 and y(t) = t3 so x3 = y2.

40. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x
x(t) = t2 and y(t) = t3 so x3 = y2.
Hence y = ±√x3

41. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x
x(t) = t2 and y(t) = t3 so x3 = y2.
Hence y = ±√x3
C(t)

42. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x
x(t) = t2 and y(t) = t3 so x3 = y2.
Hence y = ±√x3
When t = 1, C(1) = (1, 1).
C(1) = (1, 1)
C(t)

43. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x
x(t) = t2 and y(t) = t3 so x3 = y2.
Hence y = ±√x3
When t = 1, C(1) = (1, 1).
C'(t) = 2ti + 3t2j, hence C(1) = (1, 1)
C'(1) = <2, 3> is the tangent vector.
C(t)

44. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x
x(t) = t2 and y(t) = t3 so x3 = y2. C'(1) = <2, 3>
Hence y = ±√x3
When t = 1, C(1) = (1, 1).
C'(t) = 2ti + 3t2j, hence C(1) = (1, 1)
C'(1) = <2, 3> is the tangent vector.
C(t)

45. Vector-valued Functions
Algebraically, C'(t) = x'(t)i + y'(t)j + z'(t)k
y
Example E. Let C(t) = t2i + t3j. C'(t)
Sketch the curve. Find the
tangent vector at t = 1 and the
C(t)
speed at t = 1. x
x(t) = t2 and y(t) = t3 so x3 = y2. C'(1) = <2, 3>
Hence y = ±√x3
When t = 1, C(1) = (1, 1).
C'(t) = 2ti + 3t2j, hence C(1) = (1, 1)
C'(1) = <2, 3> is the tangent vector.
It's speed at t = 1 is |<2, 3>| = √13
C(t)

46. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k.

47. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k. The norm functions
of the tangent and the acceleration are
|C'(t)| = √(x'(t))2 + (y'(t))2 + (z'(t))2
|C''(t)| = √(x"(t))2 + (y"(t))2 + (z"(t))2

48. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k. The norm functions
of the tangent and the acceleration are
|C'(t)| = √(x'(t))2 + (y'(t))2 + (z'(t))2
|C''(t)| = √(x"(t))2 + (y"(t))2 + (z"(t))2
Example F. Let C(t) = ti + cos(t)j – sin(t)k, find its
tangent C'(t) and the acceleration C''(t) vector
functions.

49. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k. The norm functions
of the tangent and the acceleration are
|C'(t)| = √(x'(t))2 + (y'(t))2 + (z'(t))2
|C''(t)| = √(x"(t))2 + (y"(t))2 + (z"(t))2
Example F. Let C(t) = ti + cos(t)j – sin(t)k, find its
tangent C'(t) and the acceleration C''(t) vector
functions. Find their respective norms.

50. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k. The norm functions
of the tangent and the acceleration are
|C'(t)| = √(x'(t))2 + (y'(t))2 + (z'(t))2
|C''(t)| = √(x"(t))2 + (y"(t))2 + (z"(t))2
Example F. Let C(t) = ti + cos(t)j – sin(t)k, find its
tangent C'(t) and the acceleration C''(t) vector
functions. Find their respective norms.
The tangent function is C'(t) = i – sin(t)j – cos(t)k

51. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k. The norm functions
of the tangent and the acceleration are
|C'(t)| = √(x'(t))2 + (y'(t))2 + (z'(t))2
|C''(t)| = √(x"(t))2 + (y"(t))2 + (z"(t))2
Example F. Let C(t) = ti + cos(t)j – sin(t)k, find its
tangent C'(t) and the acceleration C''(t) vector
functions. Find their respective norms.
The tangent function is C'(t) = i – sin(t)j – cos(t)k
The acceleration function is C''(t) = – cos(t)j + sin(t)k

52. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k. The norm functions
of the tangent and the acceleration are
|C'(t)| = √(x'(t))2 + (y'(t))2 + (z'(t))2
|C''(t)| = √(x"(t))2 + (y"(t))2 + (z"(t))2
Example F. Let C(t) = ti + cos(t)j – sin(t)k, find its
tangent C'(t) and the acceleration C''(t) vector
functions. Find their respective norms.
The tangent function is C'(t) = i – sin(t)j – cos(t)k
The acceleration function is C''(t) = – cos(t)j + sin(t)k
The norm |C'(t)| = √1 + sin2(t) + cos2(t) = √2.
The norm |C''(t)| = 1.

53. Vector-valued Functions
We define the acceleration vector as
C''(t) = x''(t)i + y''(t)j + z''(t)k. The norm functions
of the tangent and the acceleration are
|C'(t)| = √(x'(t))2 + (y'(t))2 + (z'(t))2
|C''(t)| = √(x"(t))2 + (y"(t))2 + (z"(t))2
Example F. Let C(t) = ti + cos(t)j – sin(t)k, find its
tangent C'(t) and the acceleration C''(t) vector
functions. Find their respective norms.
The tangent function is C'(t) = i – sin(t)j – cos(t)k
The acceleration function is C''(t) = – cos(t)j + sin(t)k
The norm |C'(t)| = √1 + sin2(t) + cos2(t) = √2.
The norm |C''(t)| = 1.
Note in this example C' • C'' = 0 for all t, so the tangent
is always perpendicular to the acceleration for C.

54. Vector-valued Functions
We may check easily that:
1. If C(t) = <a, b, c> a constant vector, then C'(t) = 0.
2. [k*C(t)]' = k*C'(t) where k is a constant.
3. [C(t) ± D(t)]' = C'(t) ± D'(t)

55. Vector-valued Functions
We may check easily that:
1. If C(t) = <a, b, c> a constant vector, then C'(t) = 0.
2. [k*C(t)]' = k*C'(t) where k is a constant.
3. [C(t) ± D(t)]' = C'(t) ± D'(t)
There are three product rules.

56. Vector-valued Functions
We may check easily that:
1. If C(t) = <a, b, c> a constant vector, then C'(t) = 0.
2. [k*C(t)]' = k*C'(t) where k is a constant.
3. [C(t) ± D(t)]' = C'(t) ± D'(t)
There are three product rules.
Let f(t) be a function in t,
C(t) and D(t) be two vector-valued functions,
we have the following products:

57. Vector-valued Functions
We may check easily that:
1. If C(t) = <a, b, c> a constant vector, then C'(t) = 0.
2. [k*C(t)]' = k*C'(t) where k is a constant.
3. [C(t) ± D(t)]' = C'(t) ± D'(t)
There are three product rules.
Let f(t) be a function in t,
C(t) and D(t) be two vector-valued functions,
we have the following products:
* the scalar product f(t)*C(t),

58. Vector-valued Functions
We may check easily that:
1. If C(t) = <a, b, c> a constant vector, then C'(t) = 0.
2. [k*C(t)]' = k*C'(t) where k is a constant.
3. [C(t) ± D(t)]' = C'(t) ± D'(t)
There are three product rules.
Let f(t) be a function in t,
C(t) and D(t) be two vector-valued functions,
we have the following products:
* the scalar product f(t)*C(t),
* the dot product C(t)•D(t) with real number output,

59. Vector-valued Functions
We may check easily that:
1. If C(t) = <a, b, c> a constant vector, then C'(t) = 0.
2. [k*C(t)]' = k*C'(t) where k is a constant.
3. [C(t) ± D(t)]' = C'(t) ± D'(t)
There are three product rules.
Let f(t) be a function in t,
C(t) and D(t) be two vector-valued functions,
we have the following products:
* the scalar product f(t)*C(t),
* the dot product C(t)•D(t) with real number output,
* the cross-product C(t) x D(t) with vector output

60. Vector-valued Functions
Product Rules

61. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)

62. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)

63. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)
3. [C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)

64. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)
3. [C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
(For specific problems, these rules may or may not
make the calculation easier)

65. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)
3. [C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
(For specific problems, these rules may or may not
make the calculation easier)
Example G. Let C(t) = <2t, 3, t+1>,
D(t) = <t – 2, –t + 3, 2>, find [C(t) x D(t)]'.

66. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)
3. [C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
(For specific problems, these rules may or may not
make the calculation easier)
Example G. Let C(t) = <2t, 3, t+1>,
D(t) = <t – 2, –t + 3, 2>, find [C(t) x D(t)]'.
C'(t) = <2, 0, 1>, D'(t) = <1, –1, 0>

67. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)
3. [C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
(For specific problems, these rules may or may not
make the calculation easier)
Example G. Let C(t) = <2t, 3, t+1>,
D(t) = <t – 2, –t + 3, 2>, find [C(t) x D(t)]'.
C'(t) = <2, 0, 1>, D'(t) = <1, –1, 0>
[C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)

68. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)
3. [C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
(For specific problems, these rules may or may not
make the calculation easier)
Example G. Let C(t) = <2t, 3, t+1>,
D(t) = <t – 2, –t + 3, 2>, find [C(t) x D(t)]'.
C'(t) = <2, 0, 1>, D'(t) = <1, –1, 0>
[C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
= (t – 3)i – (6 – t)j + (6 – 2t)k
+ (–1 – t)i – (t +1)j + (–2t – 3)k

69. Vector-valued Functions
Product Rules
1. [f(t)*C(t)]' = f '(t)*C(t) + f(t)*C'(t)
2. [C(t)•D(t)]' = C'(t)•D(t) + C(t)•D'(t)
3. [C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
(For specific problems, these rules may or may not
make the calculation easier)
Example G. Let C(t) = <2t, 3, t+1>,
D(t) = <t – 2, –t + 3, 2>, find [C(t) x D(t)]'.
C'(t) = <2, 0, 1>, D'(t) = <1, –1, 0>
[C(t) x D(t)]' = C'(t) x D(t) + C(t) x D'(t)
= (t – 3)i – (6 – t)j + (6 – 2t)k
+ (–1 – t)i – (t +1)j + (–2t – 3)k
= –4i – 7j + (3 – 4t)k

70. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).

71. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).
A curve with constant norm
in R2 and its tangent

72. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).
y
x
A curve with constant norm A curve with constant norm
in R2 and its tangent in R3 and its tangent

73. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).
y
x
A curve with constant norm A curve with constant norm
in R2 and its tangent in R3 and its tangent

74. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).
y
x
A curve with constant norm A curve with constant norm
in R2 and its tangent in R3 and its tangent

75. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).
y
x
A curve with constant norm A curve with constant norm
in R2 and its tangent in R3 and its tangent
Proof:
|C(t)| = k means C(t)•C(t) = k2.

76. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).
y
x
A curve with constant norm A curve with constant norm
in R2 and its tangent in R3 and its tangent
Proof:
|C(t)| = k means C(t)•C(t) = k2. Differentiate both
sides, we get C(t)•C'(t) + C'(t)•C(t) = 2C(t)•C'(t) = 0.

77. Vector-valued Functions
Theorem: Let C(t) be a curve where |C(t)| = k, a
constant, then C'(t) the is always perpendicular to
C(t).
y
x
A curve with constant norm A curve with constant norm
in R2 and its tangent in R3 and its tangent
Proof:
|C(t)| = k means C(t)•C(t) = k2. Differentiate both
sides, we get C(t)•C'(t) + C'(t)•C(t) = 2C(t)•C'(t) = 0.
So C(t)•C'(t) = 0 and C(t), C'(t) are perpendicular.

78. Vector-valued Functions
Let C(t) = x(t)i + y(t)j + z(t)k be a vector-valued
function. We define
∫ C (t )dt = (∫ x(t )dt )i + ( ∫ y (t )dt ) j + ( ∫ z (t )dt )k
b b b b
∫ C (t )dt = (∫
a a
x(t ) dt )i + ( ∫ y (t ) dt ) j + ( ∫ z (t )dt ) k
a a

79. Vector-valued Functions
Let C(t) = x(t)i + y(t)j + z(t)k be a vector-valued
function. We define
∫ C (t )dt = (∫ x(t )dt )i + ( ∫ y (t )dt ) j + ( ∫ z (t )dt )k
b b b b
∫ C (t )dt = (∫
a a
x(t ) dt )i + ( ∫ y (t ) dt ) j + ( ∫ z (t )dt ) k
a a
Suppose the derivative of R(t), R'(t) = C(t), then
∫ C (t )dt = R(t ) + v, where v is a constant vector and
b
∫ C (t )dt = R(b) − R(a)
a

80. Vector-valued Functions
Let C(t) = x(t)i + y(t)j + z(t)k be a vector-valued
function. We define
∫ C (t )dt = (∫ x(t )dt )i + ( ∫ y (t )dt ) j + ( ∫ z (t )dt )k
b b b b
∫ C (t )dt = (∫
a a
x(t ) dt )i + ( ∫ y (t ) dt ) j + ( ∫ z (t )dt ) k
a a
Suppose the derivative of R(t), R'(t) = C(t), then
∫ C (t )dt = R(t ) + v, where v is a constant vector and
b
∫ C (t )dt = R(b) − R(a)
a
Since the integral is defined component-wise, all
the rules for integration of real functions are valid
for integrals of vector valued functions.

81. Vector-valued Functions
Example: Let C(t) = sin(t)i + cos(t)j + tk. Find
π
∫ C (t )dt and ∫
0
C (t )dt

82. Vector-valued Functions
Example: Let C(t) = sin(t)i + cos(t)j + tk. Find
π
∫ C (t )dt and ∫
0
C (t )dt
∫ C (t )dt = -cos(t)i + sin(t)j + ½ t2 k

83. Vector-valued Functions
Example: Let C(t) = sin(t)i + cos(t)j + tk. Find
π
∫ C (t )dt and ∫0
C (t )dt
∫ C (t )dt = -cos(t)i + sin(t)j + ½ t2 k
π 1
∫
0
C (t )dt = − cos(t )i + sin(t ) j + t 2 k |π
2
0

84. Vector-valued Functions
Example: Let C(t) = sin(t)i + cos(t)j + tk. Find
π
∫ C (t )dt and ∫0
C (t )dt
∫ C (t )dt = -cos(t)i + sin(t)j + ½ t2 k
π 1 2 π
∫0 C (t )dt = − cos(t )i + sin(t ) j + 2 t k |0
π2 π2
= (i + k ) − (−i ) = 2i + k
2 2