Physics
Normal and Binormal Vectors
Normal and binormal vectors are two important vectors in physics that are used to describe the motion of an object in three-dimensional space. The normal vector is perpendicular to the tangent vector, while the binormal vector is perpendicular to both the tangent and normal vectors. These vectors are useful in understanding the curvature and torsion of a curve.
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3 Key excerpts on "Normal and Binormal Vectors"
eBook - PDFAnton's Calculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
This will ensure that T(t) is actually tan- gent to the graph of r(t) and not sim- ply parallel to the tangent line. 12.4 Unit Tangent, Normal, and Binormal Vectors 769 Example 1 Find the unit tangent vector to the graph of r(t) = t 2 i + t 3 j at the point where t = 2. Solution. Since r ′ (t) = 2ti + 3t 2 j we obtain T(2) = r ′ (2) ‖r ′ (2)‖ = 4i + 12 j √ 160 = 4i + 12 j 4 √ 10 = 1 √ 10 i + 3 √ 10 j The graph of r(t) and the vector T(2) are shown in Figure 12.4.2. r(t) T(t) C x y Figure 12.4.1 8 10 x y (4, 8) r(t) = t 2 i + t 3 j T(2) = i + j 1 √10 3 √10 Figure 12.4.2 UNIT NORMAL VECTORS Recall from Theorem 12.2.8 that if a vector-valued function r(t) has constant norm, then r(t) and r ′ (t) are orthogonal vectors. In particular, T(t) has constant norm 1, so T(t) and T ′ (t) are orthogonal vectors. This implies that T ′ (t) is perpendicular to the tangent line to C at t, so we say that T ′ (t) is normal to C at t. It follows that if T ′ (t) ≠ 0, and if we normalize T ′ (t), then we obtain a unit vector N(t) = T ′ (t) ‖T ′ (t)‖ (2) that is normal to C and points in the same direction as T ′ (t). We call N(t) the principal unit normal vector to C at t, or more simply, the unit normal vector. Observe that the unit normal vector is defined only at points where T ′ (t) ≠ 0. Unless stated otherwise, we will assume that this condition is satisfied. In particular, this excludes straight lines. REMARK In 2-space there are two unit vectors that are orthogonal to T(t), and in 3-space there are infinitely many such vectors (Figure 12.4.3). In both cases the principal unit normal is that particular normal that points in the direction of T ′ (t). After the next example we will show that for a nonlinear parametric curve in 2-space the principal unit normal is the one that points “inward” toward the concave side of the curve. x y C C T(t) T(t) y z x There are two unit vectors orthogonal to T(t). There are infinitely many unit vectors orthogonal to T(t).
eBook - PDFCalculus
Late Transcendental
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
At each point on a smooth parametric curve C in 772 Chapter 12 / Vector-Valued Functions 3-space, these vectors determine three mutually perpendicular planes that pass through the point—the TB-plane (called the rectifying plane), the TN-plane (called the osculat- ing plane), and the NB-plane (called the normal plane) (Figure 12.4.10). Moreover, one can show that a coordinate system determined by T(t), N(t), and B(t) is right-handed in the sense that each of these vectors is related to the other two by the right-hand rule (Fig- ure 12.4.11): B(t) = T(t) × N(t), N(t) = B(t) × T(t), T(t) = N(t) × B(t) (10) The coordinate system determined by T(t), N(t), and B(t) is called the TNB-frame or some- times the Frenet frame in honor of the French mathematician Jean Frédéric Frenet (1816– 1900) who pioneered its application to the study of space curves. Typically, the xyz- coordinate system determined by the unit vectors i, j, and k remains fixed, whereas the TNB-frame changes as its origin moves along the curve C (Figure 12.4.12). Figure 12.4.10 Figure 12.4.11 Figure 12.4.12 Formula (9) expresses B(t) in terms of T(t) and N(t). Alternatively, the binormal B(t) can be expressed directly in terms of r(t) as B(t) = r (t) × r (t) r (t) × r (t) (11) and in the case where the parameter is arc length it can be expressed in terms of r(s) as B(s) = r (s) × r (s) r (s) (12) We omit the proof. QUICK CHECK EXERCISES 12.4 (See page 773 for answers.) 1. If C is the graph of a smooth vector-valued function r(t), then the unit tangent, unit normal, and binormal to C at t are defined, respectively, by T(t) = , N(t) = , B(t) = 2. If C is the graph of a smooth vector-valued function r(s) parametrized by arc length, then the definitions of the unit tangent and unit normal to C at s simplify, respectively, to T(s) = and N(s) = 3.
eBook - PDF- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
So the path of the projectile is part of a parabola. Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 958 CHAPTER 13 Vector Functions If we differentiate both sides of this equation with respect to t, we get 5 a - v9 - v9 T 1 v T9 If we use the expression for the curvature given by Equation 13.3.9, then we have 6 - | T9 | | r9 | - | T9 | v so | T9 | - v The unit normal vector was defined in Section 13.3 as N - T9 y | T9 | , so (6) gives T9 - | T9 | N - v N and Equation 5 becomes 7 a - v9 T 1 v 2 N Writing a T and a N for the tangential and normal components of acceleration, we have a - a T T 1 a N N where 8 a T - v9 and a N - v 2 This resolution is illustrated in Figure 7. Let’s look at what Formula 7 says. The first thing to notice is that the binormal vector B is absent. No matter how an object moves through space, its acceleration always lies in the plane of T and N (the osculating plane). (Recall that T gives the direction of motion and N points in the direction the curve is turning.) Next we notice that the tangential component of acceleration is v9, the rate of change of speed, and the normal component of acceleration is v 2 , the curvature times the square of the speed. This makes sense if we think of a passenger in a car—a sharp turn in a road means a large value of the curvature , so the component of the acceleration perpendicular to the motion is large and the pas- senger is thrown against the car door. High speed around the turn has the same effect; in fact, if you double your speed, a N is increased by a factor of 4.
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