Transcript Slide 1
Copyright © 2011 Pearson, Inc. Goal: Apply the arithmetic of vectors. 6.1 Day 1 Vectors in the Plane Copyright © 2011 Pearson, Inc. What you’ll learn about Two-Dimensional Vectors Vector Operations Direction Angles Applications of Vectors … and why These topics are important in many real-world applications, such as calculating the effect of the wind on an airplane’s path. Copyright © 2011 Pearson, Inc. Slide 6.1 - 3 One vs. Two Quantities Magnitude (Size) temperature distance Speed mass Copyright © 2011 Pearson, Inc. Magnitude & Direction force velocity weight Slide 6.1 - 4 Vector - magnitude: direction: Copyright © 2011 Pearson, Inc. Slide 6.1 - 5 Vocabulary Component Form: Components: Standard representation: Zero vector: Copyright © 2011 Pearson, Inc. Slide 6.1 - 6 Initial Point, Terminal Point, Equivalent Copyright © 2011 Pearson, Inc. Slide 6.1 - 7 Head Minus Tail (HMT) Rule If an arrow has initial point x1 , y1 and term inal point x 2 , y 2 , it represents the vector Copyright © 2011 Pearson, Inc. x 2 x1 , y 2 y1 . Slide 6.1 - 8 Example 1: Showing Vectors are Equivalent Show that the arrow from R = (-4, 2) to S = (-1, 6) is equivalent to the arrow from P = (2, -1) to Q = (5, 3). Copyright © 2011 Pearson, Inc. Slide 6.1 - 9 Magnitude If v is rep resen ted b y th e arro w fro m x , y to x 1 1 2 , y2 , th en v x 2 x1 y 2 2 y1 If v a , b , th en v Copyright © 2011 Pearson, Inc. . 2 a b . 2 2 Slide 6.1 - 10 Example 2: Finding Magnitude of a Vector Find the m agnitude of v represented by P Q , w here P (3, 4 ) and Q (5, 2 ). Copyright © 2011 Pearson, Inc. Slide 6.1 - 11 Vector Addition Copyright © 2011 Pearson, Inc. Slide 6.1 - 12 Vector Addition Copyright © 2011 Pearson, Inc. Slide 6.1 - 13 Example 3: Performing Vector Addition Let u = −1, 3 and v = 4,2 . Find u + v. Copyright © 2011 Pearson, Inc. Slide 6.1 - 14 Parallelogram Representation Copyright © 2011 Pearson, Inc. Slide 6.1 - 15 Scalar Multiplication scalar: Copyright © 2011 Pearson, Inc. Slide 6.1 - 16 Example 3: Performing Vector Operations Let u = −1, 2 and v = 3,2 . Find 2u. 2u - v Copyright © 2011 Pearson, Inc. Slide 6.1 - 17 Exit Ticket Performing Vector Operations Let u 2, 1 and v 5, 3 . Find 3u v . Copyright © 2011 Pearson, Inc. Slide 6.1 - 18 Goal: Use vectors to solve real-world problems. 6.1 Day 2 Vectors in the Plane Copyright © 2011 Pearson, Inc. Resolving the Vector- Copyright © 2011 Pearson, Inc. Slide 6.1 - 20 Example 5a: Finding the Components of a Vector Find the components of the vector v with direction angle 115˚ and magnitude 6. Copyright © 2011 Pearson, Inc. Slide 6.1 - 21 Example 5b: Finding the Components of a Vector Find the exact components of the vector v with direction angle 30˚ and magnitude 8. Copyright © 2011 Pearson, Inc. Slide 6.1 - 22 Example 5c: Finding the Components of a Vector Draw the indicated vector and show the components into which it is resolved. A cannonball is launched with a speed of 170 m/s at 40° above the horizontal. Copyright © 2011 Pearson, Inc. Slide 6.1 - 23 Velocity and Speed The velocity of a moving object is a vector because velocity has both magnitude and direction. The magnitude of velocity is ________. ________________ - the angle that a line of travel makes with due north, measured clockwise Copyright © 2011 Pearson, Inc. Slide 6.1 - 24 Example 7: Writing Velocity as a Vector A DC-10 jet aircraft is flying on a bearing of 65˚ at 500 mph. Find the component form of the velocity of the airplane. Recall that the bearing is the angle that the line of travel makes with due north, measured clockwise. Copyright © 2011 Pearson, Inc. Slide 6.1 - 25 Calculate Magnitude and Direction. Calculate the magnitude and direction of the vector. Copyright © 2011 Pearson, Inc. Slide 6.1 - 26 Example: Find the magnitude and direction angle of each vector. a) 12, 4 Copyright © 2011 Pearson, Inc. b) Slide 6.1 - 27 Summary Magnitude and direction → 𝑎, 𝑏 𝑎, 𝑏 → magnitude and direction Copyright © 2011 Pearson, Inc. Slide 6.1 - 28 Goal: Calculate the resultant vector. 6.1 Day 3 Vectors in the Plane Copyright © 2011 Pearson, Inc. Resultant Vector Copyright © 2011 Pearson, Inc. Slide 6.1 - 30 Example: Add the Vectors to find the Resultant Vector Copyright © 2011 Pearson, Inc. Slide 6.1 - 31 Example: Calculating the Effect of Wind Velocity A jet carrying Dora the Explorer is flying at 400 mph on a course with a bearing of 30º. If the jet experiences a crosswind blowing due south at 20 mph, find the resultant speed and direction of the jet. Round all values throughout the problem and the final answer to the nearest tenth. Copyright © 2011 Pearson, Inc. Slide 6.1 - 32 : Combining Forces Copyright © 2011 Pearson, Inc. Slide 6.1 - 33 Solution Copyright © 2011 Pearson, Inc. Slide 6.1 - 34