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CHAPTER 2. VECTOR FUNCTIONS
2.4
Tangent, Normal and Binormal Vectors
Three vectors play an important role when studying the motion of an object
along a space curve. These vectors are the unit tangent vector, the principal normal vector and the binormal vector. We have already de…ned the unit tangent
vector. In this section, we de…ne the other two vectors.
Let us start by reviewing the de…nition of the unit tangent vector.
De…nition 147 (Unit Tangent Vector) Let C be a smooth curve with posi!
tion vector !
r (t). The unit tangent vector, denoted T (t) is de…ned to be
!
!
r 0 (t)
T (t) = !0
k r (t)k
2.4.1
Normal and Binormal Vectors
De…nition 148 (Normal Vector) Let C be a smooth curve with position vector !
r (t). The principal normal vector or simply the normal vector, denoted
!
N (t) is de…ned to be:
!0
!
T (t)
N (t) = !
(2.13)
T 0 (t)
The name of this vector suggests that it is normal to something, the question
!
!
is to what? By de…nition, T is a unit vector, that is T (t) = 1. From
!
!
!
!
proposition 120, it follows that T 0 (t) ? T (t). Thus, N (t) ? T (t). In fact,
!
!
N (t) is a unit vector, perpendicular to T pointing in the direction where the
curve is bending.
De…nition 149 (Binormal Vector) Let C be a smooth curve with position
!
vector !
r (t). The binormal vector, denoted B (t), is de…ned to be
!
!
B (t) = T (t)
!
N (t)
!
!
Since both T (t) and N (t) are unit vectors and perpendicular, it follows that
!
!
!
B (t) is also a unit vector. It is perpendicular to both T (t) and N (t).
Example 150 Consider the circular helix !
r (t) = hcos t; sin t; ti. Find the unit
tangent, normal and binormal vectors.
!
!
r 0 (t)
Unit Tangent: Since T (t) = !0
, we need to compute !
r 0 (t) and
k r (t)k
k!
r 0 (t)k.
!
r 0 (t) = h sin t; cos t; 1i
2.4. TANGENT, NORMAL AND BINORMAL VECTORS
91
and
!
r 0 (t)
=
=
Thus
p
p
!
T (t) =
sin2 t + cos2 t + 1
2
sin t cos t 1
p ; p ;p
2
2
2
!0
!
!
!
T (t)
, we need to compute T 0 (t) and T 0 (t) .
Normal: Since N (t) = !
T 0 (t)
!0
T (t) =
sin t
cos t
p ; p ;0
2
2
and
!0
T (t)
=
=
Thus
s
cos2 t sin2 t
+
2
2
1
p
2
!
N (t) = h cos t;
sin t; 0i
Binormal:
!
B (t)
!
!
T (t) N (t)
sin t cos t 1
p ; p ;p
=
2
2
2
!
sin t
cos t 1
p ; p ;p
B (t) =
2
2
2
=
h cos t;
sin t; 0i
The pictures below (…gures 2.5, 2.6 and 2.7) show the helix for t 2 [0; 2 ]
!
!
!
as well as the three vectors T (t), N (t) and B (t) plotted for various
values of t. If the three vectors do not appear to be exactly orthogonal, it
is because the scale is not the same in the x; y and z directions.
2.4.2
Osculating and Normal Planes
De…nition 151 (Osculating and Normal Planes) Let C be a smooth curve
with position vector !
r (t). Let P be a point on the curve corresponding to !
r (t0 )
for some value of t.
!
!
1. The plane through P determined by N (t0 ) and B (t0 ) is called the normal
!
plane of C at P . Note that its normal will be T (t).
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CHAPTER 2. VECTOR FUNCTIONS
!
!
!
Figure 2.5: Helix and the vectors T (0), N (0) and B (0)
!
!
!
Figure 2.6: Helix and the vectors T (1), N (1) and B (1)
2.4. TANGENT, NORMAL AND BINORMAL VECTORS
93
!
!
!
Figure 2.7: Helix and the vectors T (4), N (4) and B (4)
!
!
2. The plane through P determined by T (t0 ) and N (t0 ) is called the oscu!
lating plane of C at P . Note that its normal will be B (t).
Example 152 Find the normal and osculating planes to the helix given by
!
.
r (t) = hcos t; sin t; ti at the point 0; 1;
2
Earlier, we found that
!
T (t) =
sin t cos t 1
p ; p ;p
2
2
2
!
N (t) = h cos t;
sin t; 0i
and
!
B (t) =
At the point 0; 1;
2
, that is when t =
!
T
=
2
!
N
cos t 1
p ;p
2
2
sin t
p ;
2
2
2
, we have
1
1
p ; 0; p
2
2
= h0; 1; 0i
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CHAPTER 2. VECTOR FUNCTIONS
and
!
B
2
1
1
p ; 0; p
2
2
=
Normal Plane: It is the plane through
0; 1;
1
1
p ; 0; p . Thus its equation is
2
2
1
1
p (x 0) + p z
2
2
p
Multiplying each side by 2 gives
(x
0) + z
2
=
=0
2
2
!
with normal T
2
=0
or
z
x=
2
Osculating Plane: It is the plane through 0; 1;
1
1
p ; 0; p
2
2
!
with normal B
2
2
=
Thus its equation is
1
1
p (x 0) + p z
2
2
p
Multiplying each side by 2 gives
(x
0) + z
2
2
=0
=0
or
x+z =
2
Make sure you have read, studied and understood what was done above
before attempting the problems.
2.4.3
Problems
Do # 11, 13, 37, 39, 43 at the end of 10.3 in your book