6.3 Vectors Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 Students will: Represent vectors as directed line segments. Write the component form of vectors. Perform basic vector operations and represent vectors graphically. Write vectors as linear combinations of unit vectors. Find the direction angles of vectors. Use vectors to model and solve real-life problems.
Do you remember? Which Trig Law should you use with each given information:(Law of Sines/Cosines?) SASASA SSSSSA AAS Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
3 A ball flies through the air at a certain speed and in a particular direction. The speed and direction are both important quantities of the ball. The velocity is a vector quantity since it has both a magnitude and a direction. Vectors are used to represent velocity, force, tension, and many other quantities. A vector is a quantity with both a magnitude and a direction. Definition: Vector
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Directed Line Segment A quantity with magnitude and direction is represented by a directed line segment PQ. Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude. And are in the same direction. u v P Q Initial point Terminal point The vector v = PQ is the set of all directed line segments of length ||PQ|| which are parallel to PQ.
The Distance Formula: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
6 Example 1 Equivalent Directed Line Segments Let u be the directed line segment from P=(0,0) to Q=(3,2) Let v be the directed line segment from R=(1,2) to S=(4,4) Show that u = v (Check magnitude and slope)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (v 1, v 2 ). Standard Position x y (v 1, v 2 ) Components of v P Q If point P is the origin and point Q = (5, 6), then v = The component form of vector v is written
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Definition: Component Form and Magnitude If v is a vector with initial point P = (p 1, p 2 ) and terminal point Q = (q 1, q 2 ), then x y 1. The component form of v is 2. The magnitude (or length) of v is P Q (p 1, p 2 ) (q 1, q 2 )
Try #1 page 417
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Component Form and Magnitude Example 2a: Find the component form and magnitude of vector v that has initial point (2, 2) and terminal point (–1, 4). The magnitude of v is x y P = (2, 2) Q = (–1, 4) The component form of v is 3.61 units
Example 2b Finding the Component Form of a Vector Find the component form and magnitude of the vector v that has initial point (4,7) and terminal point (-1,5)
Try #11 page 417 Initial pt. (-3,-5) Terminal pt. (5,1)
Scalar Multiplication Two basic vector operations are scalar multiplication and vector addition. Scalar multiplication is the product of ku=k =
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Scalar multiplication is the product of a scalar, or real number, times a vector. Scalar Multiplication Example: Given vector find –2u. x y 4 -4 –2u The product of –2 and u gives a vector twice as long as and in the opposite direction of u. u
Vector Addition To add two vectors position them (without changing their length or direction) Position the vectors so that the initial point of one is located at the terminal point of the other.This technique is called the parallelogram law for vector addition The vector u + v is often called the resultant of vector addition.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Vector Addition and Subtraction Example: Given vectors find u + v and u – v. Example 3a Vector Addition and Subtraction
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Try #19 page 418 u =, v = a. u+v b. u-v c. 2u-3v d. v+4u
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Unit Vectors A unit vector in the direction of v is a vector, u, that has a magnitude of 1 and the same direction as v. Divide v by its length to obtain a unit vector. Example: Find a unit vector in the direction of v = –1, 1 . unit vector in the direction of v
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Example #4 Finding a Unit Vector Find a unit vector in the direction of v = Verify that the result has a magnitude of 1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Try #28 page 418
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Standard Unit Vectors The unit vectors and are called standard unit vectors and are denoted by i = and j = * NOTE: The letter i is written in bold to distinguish the difference it from imaginary i. The scalars and are the horizontal and vertical components of v The linear combination of the vectors i and j is i + j
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Example #5 Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2,5) and terminal point (-1,3). Write u as a linear combination of the standard vectors i and j.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Example #6 Vector Operations Let u = -3i + 8j and v = 2i – j. Find 2u – 3v
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Direction Angles If u is a unit vector such that is the angle (measured counterclockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and is the direction angle of the vector u. y x 1 1 –1–1 – 1 (x, y) y = sin x = cos u Continued.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 Direction Angles Direction Angles continued If v = ai + bj is any vector that makes an angle with the positive x-axis, then it has the same direction as u and The direction angle for v is determined by Example: The direction angle of u = 3i +3j is
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 Example 7 Finding Direction Angles of Vectors Find the direction angle of each vector a.) 3i + 3j b.) 3i – 4j
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Try #55 page 418
Homework Day 1 p #1-3,5,11-12,15-16,19-20,28-29 Day 2 p #41-42,47-48,51-52,54-55,63,67, 69,73,75,81,82 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28