Calculus 1 for AE (WI1421LR) Lecture 1: 12.3 The dot product 12.4 The cross product

Transcription

1 Calculus 1 for AE (WI1421LR) : 12.3 The dot product 12.4 The cross product

2 About this course Textbook James Stewart, Calculus, Early Transcendentals, 7th edition Course Coordinator Dr. Roelof Koekoek, Department of Applied Mathematics Course documents en.html About me (instructor) Juan Juan Cai: Office: HB , EWI

3 Working schedule Calculus I for AE (wi1421lr), semester 1A James Stewart, Calculus, Early Transcendentals, 7th edition, Brooks/Cole Cengage Learning, 2012 week subjects minimum extra remarks 1 ( 12.1), ( 12.2) : 3, 5, 9, 15, 19, 21, 22, 27, 45, : 3, 5, 7, 17, 19, 27, 31, 37, 43, : 1, 2, 7, 10, 17, 25, 56, 63, : 1, 2, 6, 13, 21, 23, 28, 30, 34, 38, 50 pages : 63, 65, 67, 69, : 64(a), 66(b), 68(b), 70, 72 2 ( 2.2), : 11, 13, 17, 21, : 15, 19, 23, 26, 28 only arcsin, arccos and arctan ( 2.5), : 15, 19, 21, 25, : 17, 23, 26, 27, 37 ( 3.4) 3.4 : 7, 11, 17, : 8, 12, 20, : 1, 3, 7, 15, 29, 35, 51, : 27, 37, 49, 54, : 1, 5, 13, 15, 28, : 3, 11, 17, 26, ( 5.2), ( 5.3) : 9, 11, 17, 23, 25, 35, : 3, 5, 13, 21, 25, 40, 45, 53, 59, 70, : 7, 13, 19, 21, 29, 33, : 1, 4, 7, 20, 30, 33, 38, 44, 66, : 1, 3, 15, 23, 39, 49, : 5, 17, 24, 33, 38, 42, 48, 52 5 ( 7.5) 7.5 : 1, 7, 17, 21, 31, : 9, 23, 42, 43, : 1, 5, 13, 21, 31, 49, 50, 55, : 2, 11, 17, 25, 52, 58, 69, 75 ( 9.1) 9.1 : 1, 3, : 2, 4, : 3, 9, 11, 45, : 1, 5, 12, 46, 48 skip orthogonal trajectories : 3, 7, 9, 13, 17, : 1, 5, 8, 19, 34, 35 7 App. H App. H : 3, 5, 8, 11, 15, 19, 23, 25, 27, 29 App. H : 33, 35, 37, 39, 41, 43, 45, 47 App. H : 1, 9, 12, 16, 17, 20, 21, 22, 26, 28, 31 App. H : 34, 36, 38, 40, 42, 44, 46, 48

4 Introduction on vectors A vector is determined by its magnitude and direction. The two vectors u and v are equivalent: u = v. In 2-dimensional rectangular coordinate system, a :=< a 1, a 2 > denotes the vector initiating at origin O and pointing toward the point P (a 1, a 2 ). The coordinates a 1 and a 2 are called the components of a. A three dimensional vector can be defined similarly.

5 Algebraic operations on vectors Let a :=< a 1, a 2 >, b :=< b 1, b 2 > and c be a real number. Then, a + b =< a 1 + b 1, a 2 + b 2 > a b =< a 1 b 1, a 2 b 2 >. The scalar multiplication of a vector is given by ca =< ca 1, ca 2 >. The operations on three-dimensional vectors are defined in the same way. Triangle Law & Parallelogram Law:

6 Standard basis vectors Two-Dim: i =< 1, 0 > and j =< 0, 1 >. Three-Dim: i =< 1, 0, 0 > and j =< 0, 1, 0 > and k =< 0, 0, 1 >. For a vector a =< a 1, a 2, a 3 >, we can write a = a 1 i + a 2 j + a 3 k. The length of the vector a is denoted by the symbol a or a. < a 1, a 2 > = a a2 2 < a 1, a 2, a 3 > = a a2 2 + a2 3 A vector with length 1 is called unit vector. For instance, the standard basis vectors are unit vectors. The vectors 0 =< 0, 0 > and 0 =< 0, 0, 0 > are called zero vectors.

7 Dot product If a =< a 1, a 2, a 3 > and b =< b 1, b 2, b 3 >, then the dot product of a and b is defined as a b = a 1 b 1 + a 2 b 2 + a 3 b 3. If a =< a 1, a 2 > and b =< b 1, b 2 >, then a b = a 1 b 1 + a 2 b 2. Note that dot product of two vectors is a real number. If a, b and c are vectors and c is a scalar, then 1 a a = a 2 2 a b = b a 3 a (b + c) = a b + a c 4 (ca) b = c(a b) = a (cb) 5 0 a = 0

8 Angles between vectors If θ is the angle between the vectors a and b, then a b = a b cos θ. If vectors a and b both have positive length, then cos θ = a b a b. The angle remains the same while changing the lengths of a and b. Two vectors a and b are orthogonal if and only if a b = 0.

9 Examples If the vectors a and b have lengths 4 and 6, and the angle between them is π/3, find a b. Find the angles between the vectors a =< 2, 2, 1 > and b =< 5, 3, 2 >. Show that 2i + 2j k is perpendicular to 5i 4j + 2k

10 The direction angles of a nonzero vector a are the angles between a and the positive axes, denoted by α, β, and γ. Since the basis vector i represents the direction of positive x-axis, Similarly, we have cos β = a j a j = a 2 a cos α = a i a i = a 1 a. cos γ = a k a k = a 3 a. Hence, it is easy to prove that cos 2 α + cos 2 β + cos 2 γ = 1.

11 Projections Vector projection of b onto a: proj a b = ( a b a 2 ) a. Scalar projection of b onto a: comp a b = a b a = b cos θ. The vector projection is a vector. You can think it as the shadow of b when the light is perpendicular to a. The scalar projection is a real number, the absolute value of which is the length of proj a b.

12 Example Find the scalar projection and vector projection of b =< 1, 1, 2 > onto a =< 2, 3, 1 >.

13 The cross product If a =< a 1, a 2, a 3 > and b =< b 1, b 2, b 3 >, then the cross product of a and b is the vector a b =< a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 >. Note that the cross product is NOT defined for two dimension vectors. The vector a b is orthogonal to both a and b. The length of the cross product a b is equal to the area of the parallelogram determined by a and b: a b = a b sin θ. Two nonzero vectors a and b are parallel if and only if a b = 0.

14 If a, b and c are vectors and c is a scalar, then 1 a b = b a 2 (ca) b = c(a b) = a (cb) 3 a (b + c) = a b + a c 4 (a + b) c = a c + b c 5 a (b c) = (a b) c 6 a (b c) = (a c)b (a b)c The vector a (b c) is called the scalar triple product, the absolute value of which equals to the volume of the parallelepiped determined by the vectors a, b and c.

15 Examples Find a vector perpendicular to the plane that passes through the points P (1, 4, 6), Q( 2, 5, 1), and R(1, 1, 1). Find the area of the triangle with vertices P (1, 4, 6), Q( 2, 5, 1), and R(1, 1, 1). Use the scalar triple product to show that the vectors a =< 1, 4, 7 >, b =< 2, 1, 4 >, and c =< 0, 9, 18 > are coplanar.