Vectors in the Plane Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Objective Represent vectors as directed line segments. Write the component forms of vectors. Perform basic vector operations. Write vectors as linear combinations. Find direction angles of vectors.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity 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. Magnitude and Direction

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 A quantity with magnitude and direction is represented by a directed line segment PQ with initial point P and terminal point Q. Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude. u v Directed Line Segment The vector v = PQ is the set of all directed line segments of length ||PQ|| which are parallel to PQ. P Q

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Vectors have both magnitude and direction. Magnitude = length. (also called modulus) Donated by We can use the distance formula to find the Magnitude. Direction is determined by the angle (measured counterclockwise) the line segment makes with the positive x-axis.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example 1 Let u be represented by the directed line segment from P = (0, 0) to Q = (3, 1) and let v be represented by the directed line segment from R = (2, 2) to S = (5, 3). Show that u = v

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Component form – vector whose initial point is the origin and is uniquely represented by the coordinates of its terminal point. The coordinates are the components of v.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 The component form of the vector with initial point

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 The magnitude (or length) of v is given by If v is a unit vector. Moreover,if and only if v is the zero vector 0.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example 2 Find the component form and magnitude the vector v that has initial point (2, -3) and terminal point (-7, 9)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Equal Vectors Example 3: If u = PQ, v = RS, and w = TU with P = (1, 2), Q = (4, 3), R = (1, 1), S = (3, 2), T = (-1, -2), and U = (1, -1), determine which of u, v, and w are equal. Calculate the component form for each vector: u = 4  1, 3  2 = 3, 1 v = 3  1, 2  1 = 2, 1 w = 1  (-1),  1  (-2) = 2, 1 Therefore v = w but v = u and w = u. // Two vectors u = u 1, u 2 and v = v 1, v 2 are equal if and only if u 1 = v 1 and u 2 = v 2.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Definitions of Vector Addition and Scalar Muliplication

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Vector Addition To add vectors u and v: 1. Place the initial point of v at the terminal point of u. 2. Draw the vector with the same initial point as u and the same terminal point as v. Vector Addition v u

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Vector Subtraction To subtract vectors u and v: 1. Place the initial point of v at the initial point of u. 2. Draw the vector u  v from the terminal point of v to the terminal point of u. Vector Subtraction v u v u v u u  v

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Scalar multiplication is the product of a scalar, or real number, times a vector. For example, the scalar 3 times v results in the vector 3v, three times as long and in the same direction as v. v 3v3v v The product of - and v gives a vector half as long as and in the opposite direction to v. - v Scalar Multiplication

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Example 4

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Properties of Vector addition and Scalar Multiplication

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Unit Vectors A vector with a magnitude of 1.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Unit Vectors

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Example 5

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Unit Vectors

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Linear Combination of Unit Vectors

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Example 6

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 Example 7 Let u = i + j and v = 5i - 3j. Find 2u – 3v.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 x y The direction angle  of a vector v is the angle formed by the positive half of the x-axis and the ray along which v lies. x y v θ v θ x y v x y (x, y) Direction Angle If v = 3, 4, then tan  = and  = . If v = x, y, then tan  =.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Direction Vector The general form of the direction vector

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 Example 8 Find the direction angle of each vector. A. 3i + 3j B. 3i – 4j

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Example 9 Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 miles per hour at an angle 45 degrees below the horizontal.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 Example 10 A force of 500 pounds is required to pull a boat and trailer up a ramp inclined at 12 degrees from the horizontal. Find the combined weight of the boat and trailer.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31 Example 11 An airplane is traveling at a speed of 724 kilometers per hour at a bearing of N 30 degrees E. If the wind velocity is 32 kilometers per hour from the west find the resultant speed and direction of the plane.