Review of Coordinate Systems

Transcription

1 Vector in 2 R and 3 R

2 Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar

3 Cartesian Coordinate System Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y)

4 Polar Coordinate System Origin and reference line are noted Point is distance r from the origin in the direction of angle, from reference line The reference line is often the x-axis. Points are labeled (r,)

5 Polar to Cartesian Coordinates Based on forming a right triangle from r and x = r cos y = r sin If the Cartesian coordinates are known: tan y x r x y 2 2

6 Vectors in Two and Three Dimensions Algebraic and geometric points of view

7 Algebraic Definitions A vector in R 2 is an ordered pair of real numbers, e.g., (1, 3) or (π, 2). In general we write a = (a 1, a 2 ) for a vector in R 2. A vector in R 3 is an ordered triple of real numbers, e.g., (7,18, 9) or (e,0, 2). In general we write a = (a 1, a 2, a 3 ) for a vector in R 3.

8 Arithmetic Operation: Vector Addition Add vectors in the same space by adding componentwise. Examples (1, 3) + (5, 7) = (6, 4). ( 2, 0, 1) + ( 2, 5, 1) = ( 2 2, 5, 0). (4, 1, 8) + (1, 2) is not defined. In general, if a = (a 1, a 2, a 3 ) and b = (b 1, b 2, a 3 ), then a + b = (a 1 +b 1, a 2 +b 2, a 3 +b 3 ). (Similar for vectors in R 2.)

9 Arithmetic Operation: Scalar Multiplication Multiply a vector by a single real number (called a scalar) by multiplying the components of the vector by that real number. Examples 3(1, 7) = (3, 21). 2(1, 0, 3) = ( 2, 0, 6). In general, if a = (a 1, a 2, a 3 ) and k R, then ka = (ka 1, ka 2, ka 3 ). (Similar for vectors in R 2.)

10 Unit vectors Length / magnitude = 1 In R 2 y j i x z In R 3 k x i j y

11 a = a 1 i + a 2 j b = =b 1 i + b 2 j + b 3 j

12 Length of vectors a = b = a a b b b 1 2 3

13 Some Properties For all a, b, c R n (n = 2 or 3) and k, l R, we have: 1. a + b = b + a. 2. a + (b + c) = (a + b) + c. 3. There s a special vector 0 such that a + 0 = a. 4. k(a + b) = ka + kb. 5. (k + l)a = ka + la. 6. k(la) = (kl)a = l(ka).

14 Geometric Ideas A vector a R n (n = 2 or 3) can be represented by an arrow from the origin to the point (a 1, a 2 ) or (a 1, a 2, a 3 ). R 2 y R 3 z (a 1, a 2, a 3 ) a (a 1, a 2 ) a x x y

15 However, any parallel translate of the arrow from the origin may also serve as a representation of a. R 2 y R 3 z a a x x y Any one of the arrows shown above represents the vector a. The red arrow is referred to as the position vector of the point (a 1, a 2 ) or (a 1, a 2, a 3 ).

16 Visualization of Vector Addition Translate b so its tail is at the head of a, then add arrows tail-to-head. Or, a + b is the diagonal of the parallelogram determined by a and b. b a + b b (translated) a

17 Adding Some Vectors Graphically When you have many vectors, just keep repeating the process until all are included. The resultant is still drawn from the tail of the first vector to the head of the last vector. u s v w

18 Adding Vectors, Rules When two vectors are added, the sum is independent of the order of the addition. This is the Commutative Law of Addition. u + v = v + u v u v u

19 Adding Vectors, Rules cont. When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped. This is called the Associative Property of Addition. u + (v + w) = (u + v) + w w w u v u v

20 Visualization of Scalar Multiplication Stretch or compress a by the scalar k. Reverse the sense of the arrow if k < 0. a a (1/3)a 2a

21 Subtraction of Vectors Vector a b is a + ( b). a + b b a b a b (translated) a b is the other diagonal of the parallelogram determined by a and b. b a b

22 Displacement Vector Between Two Points The displacement vector from the point P 1 (a 1, a 2, a 3 ) to the point P 2 (b 1, b 2, b 3 ), denoted P 1 P 2, is obtained by subtracting position vectors. z P 1 P 1 P 2 P 2 P 1 P 2 = OP 2 OP 1 = (b 1, b 2, b 3 ) (a 1, a 2, a 3 ) = (b 1 a 1, b 2 a 2, b 3 a 3 ). x O y

23 The Dot Product A scalar product

24 Definition a b = a b cos, where, for 0 π, is the angle between a and b. b a

25 Theorem Let a = (a 1, a 2, a 3 ) and b = (b 1, b 2, b 3 ) and be vectors in R 3. Then the dot product a b is a b = a 1 b 1 + a 2 b 2 + a 3 b 3. (There s an analogous definition for vectors in R 2.) Example (1, 3, 2) (2, 1, 5) = 12 + ( 3) = 9.

26 where vw a a b b v a i b jand w a i b j If v 2i 5j and w 4i 1 j, find v w v w This is called the dot product. Notice the answer is just a number NOT a vector.

27 Angles between Vectors When a and b are nonzero vectors, we can find the angle between them by cos 1 a b a b. b a Hence a is perpendicular (orthogonal) to b if and only if a b = 0.

28 The dot product is useful for several things. One of the important uses is in a formula for finding the angle between two vectors that have the same initial point. If u and v are two nonzero vectors, the angle, 0 <, between u and v is determined by the formula v u cos Technically there are two angles between these vectors, one going the "shortest" way and one going around the other way. We are talking about the smaller of the two. u u v v

29 Find the angle between u = 2i jand v = 4i +3j. u v cos u v u v 2 2 u v v 4i 3j cos u u v v u 2i j cos

30 cos Find the angle between the vectors v = 3i + 2j and w = 6i + 4j vw v w cos What does it mean when the angle between the vectors is 0? v 3i 2j w 6i 4j The vectors have the same direction. We say they are parallel because remember vectors can be moved around as long as you don't change magnitude or direction.

31 If the angle between 2 vectors is cos Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal. v u u v v w, what would their dot product be? Since cos is 0, the dot product must be 0. Vectors u and v in this v case = 4i are - j called orthogonal. (similar to perpendicular but refers to vectors). compute their dot product and see if it is 0 w = 2i + 8j v w The vectors v and w are orthogonal.

32 Compute the dot product of each pair of vectors: i i i j j j

33 Given the vectors u = 8i + 8j 3k and v = 10i + 11j 2k find the following. u v v u v v

34 Given the vectors: u = 8i + 8j 3k, v = 10i +11j 2k, and w = 9i + 7j + 5j ; find the following. w v u w w u v 12 3 w u v u

35 Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i 7j.

36 Determine if the pair of vectors is orthogonal, parallel, or neither. u 8i 3j and v 2i 16 3 j u 7i 3j and v 21i 9j

37 Algebraic Properties 1. a a 0; a a = 0 if and only if a = a b = b a. 3. (ka) b = k(a b) = a (kb). 4. a (b + c) = a b + a c.

38 Connection to Length z a a 1 2 a2 2 a3 2 a a a a 3 x a 2 a 1 y

39 Normalization of a Vector If a 0, a unit vector u in the direction of a is given by u a a. u = 1

40 Projections The next slide shows representations PQ and PR of two vectors a and b with the same initial point P. If S is the foot of the perpendicular from R to the line containing PQ, then the vector with representation PS is called the vector projection of b onto a and is denoted by proj a b.

41 Projections (cont d)

42 Projections (cont d) The scalar projection of b onto a is the length b cos θ of the vector projection. This is denoted by comp a b, and can also be computed by taking the dot product of b with the unit vector in the direction of a.

43 Vector Projection Assume a 0. b a proja b a b a a a proj a b

44 The Cross Product A vector product

45 Definition a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k i j k axb a b 1 1 a b 2 2 a b 3 3. a b a 2 a 3 i a 1 a 3 j a 1 a 2 k b 2 b 3 b 1 b 3 b 2 b 1

46 Coordinate Formula a b = (a 2 b 3 a 3 b 2 )i (a 1 b 3 a 3 b 1 )j + (a 1 b 2 a 2 b 1 )k

47 Geometric Definition Let a = (a 1, a 2, a 3 ) and b = (b 1, b 2, b 3 ) be vectors in R 3. The cross product a b is the vector in R 3 with the following features:

48 The magnitude (length) of a b is the area of the parallelogram determined by a and b. b a a b = a b sin The direction of a b (if nonzero) is perpendicular to both a and b and is such that (a, b, a b) is right-handed. a b b a

49

50 Algebraic Properties 1. a b = b a 2. a (b + c) = a b + a c 3. (a + b) c = a c + b c 4. (ka) b = k(a b) = a (kb)