Unit 11: Vectors in the Plane

Transcription

1 135 Unit 11: Vectors in the Plane Vectors in the Plane The term ector is used to indicate a quantity (such as force or elocity) that has both length and direction. For instance, suppose a particle moes along a line segment from point A to point B. The directed line segment AB with initial point A, terminal point B, and length (or magnitude) AB can be used to represent this displacement. The zero ector, denoted by 0, has length 0. Directed line segments that hae the same magnitude and direction are equialent (or equal). The set of all directed line segments that are equialent to AB is a ector in the plane and is denoted by = AB. We denote a ector by a boldface letter (, u, w... ) or by putting an arrow aboe the letter (, u, w... ). Ex. 1: Let be represented by the directed line segment from (0,0) to (3,2), and let u be represented by the directed line segment from (1,2) to (4,4). Show that and u are equialent.

2 136 Definition of Vector Addition and Scalar Multiplication Vector Addition If u and are ectors positioned so that the initial point of is at the terminal point ofu, then the sum u is the ector from the initial point of u to the terminal point of. The ector u is called the resultant ector. Ex.2: Draw the sum of ectors a and b as shown. b a Scalar Multiplication If c is a scalar and is a ector, then the scalar multiple c is a ector whose length is c times the length of and whose direction is the same as if c 0 and is opposite to if c 0. If c 0 or 0, then c 0.

3 137 Ex. 3 If u and are the ectors shown, then draw the following: a. u b. 23u u Component Form of a Vector in the Plane If is a ector in the plane whose initial point is the origin (standard position) and whose terminal point (, ), then the component form of is gien by is 1 2,. 1 2 The coordinates 1 and 2 are called the components of. The component form of the zero ector is 0 0,0. Gien the points A( x1, y1) andb( x2, y 2), the ector a representing AB is x2 x1, y2 y1. Ex. 4 Graph the ector represented by the directed line segment with initial point A(2,3) and terminal point B( 2,1). Find the component form of the ector.

4 138 Using the component form: a a, a and b b1, b2 If 1 2, then the magnitude of ector a a1, a2 is a a a the sum of ectors a and b is ab a1 b1, a2 b2 the difference of ectors a and b is ab a1 b1, a2 b2 the ector a times the scalar multiple of c a is ca ca1, ca2. Ex5. If a 4,0 and b 2,1, find a. a b. a b c. ab d. 3b e. 2a 5b Properties of Vector Addition and Scalar Multiplication Let uand, w be ectors and c and d be scalars. Then the following properties are true. 1. u u 2. ( u ) w u ( w ) 3. u 0 u 4. u ( u) 0 5. c( du) ( cd) u 6. ( c d) u cu du 7. c( u ) cu c 8. 1( u) u,0( u) 0 9. c( u ) 10. c c

5 139 Ex6. Proe property #6. Ex.7 Proe property #10 Unit Vector in the Direction of If is a nonzero ector in the plane, then the ector u 1 is called the unit ector in the same direction of and has a magnitude of 1. Proof: Ex8. Find a unit ector in the direction of 2,5 and erify that it has length 1.

6 140 Standard Unit Vectors The unit ectors 1,0 and 0,1 are called the standard unit ectors in the plane and are denoted by i 1,0 and j 0,1 as shown: These ectors can be used to uniquely represent any ector as follows., i j Proof: The ector 1i 2j is called a linear combination of i and j. The scalars 1 and 2 are called the horizontal and ertical components of. Ex9. Let u be the ector with initial point (2, 5) and terminal point ( 1,3), and let 2i j. Write each of the ectors as a linear combination of i and j. a. u b. w 2u 3

7 141 If u is a unit ector such that is the angle (measured counterclockwise) from the positie x -axis to u, then the terminal point of u lies on the unit circle and you hae u cos, sin cos i sin j as shown: Moreoer, it follows that any other nonzero ector making an angle with the positie x -axis has the same direction as u, and be written as cos, sin cos i sin j Ex10. The ector has a length of 3 and makes an angle of 30 a linear combination of the unit ectors i and j. with the positie x -axis. Write as 6

8 142 Applications of Vectors There are many applications of ectors in the real world. One example is force, because force has both magnitude and direction. If two or more ectors are acting on an object, then the resultant force on the object is the sum of the ector forces. Ex11. Three forces with magnitudes of 75 pounds, 100 pounds and 125 pounds act on an object at angles 30, 45 and 120 degrees respectiely. Find the direction and magnitude of the resultant forces. Ex12. An airplane is traeling at a fixed altitude with a negligible wind factor. The plane is headed 30 W (30 degrees west of north) at a speed of 500 miles per hour. As the plane reaches a certain point, it encounters a wind with a elocity of 70 miles per hour in the direction of E 45 N. What are the resultant speed and direction of the plane?

9 143 Definition of Dot Product The dot product of u u1, u2 and 1, 2 is u Because the dot product of two ectors yields a scalar, it is also called the scalar product (or inner product) of the two ectors. Properties of the Dot Product Let uand, w be ectors in the plane and let c be a scalar. 1. u u 2. u( w) u uw 3. c( u ) cu uc Ex1. Proe property 5. Ex2. Gienu 2, 2, 5,8, and w 4,3, find each of the following. a. u b. ( u ) w c. u(2 ) d. w 2

10 144 Angle between Two Vectors The angle between two nonzero ectors in their respectie standard position is the angle, 0. If is the angle between two nonzero ectors u and, then cos u u Proof (use the Law of Cosines): Ex3. Find the angle between u 4,3 and 3,5

11 145 Rewriting the dot product in the form u u cos, we can more easily see the fie different orientations of the two ectors based on the alue of And from this, we get the following definition: The ectors u and are orthogonal if u 0. Ex3. Are the ectors u 2, 3 and 6, 4 orthogonal?