Vectors in the Plane Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 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. 3 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. 4 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. 5 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. u v u + v v u Vector Addition v u

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 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. 7 A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (u 1, u 2 ). Standard Position If v is a vector with initial point P = (p 1, p 2 ) and terminal point Q = (q 1, q 2 ), then 1. The component form of v is v = q 1  p 1, q 2  p 2 2. The magnitude (or length) of v is ||v|| = x y (u 1, u 2 ) x y P (p 1, p 2 ) Q (q 1, q 2 )

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Find the component form and magnitude of the vector v with initial point P = (3,  2) and terminal point Q = (  1, 1). Example: Magnitude The magnitude of v is ||v|| = = = 5. =, 34  p 1, p 2 = 3,  2 q 1, q 2 =  1, 1 So, v 1 =  1  3 =  4 and v 2 = 1  (  2) = 3. Therefore, the component form of v is, v 2 v1v1

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Equal Vectors Example: 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. 10 Operations on Vectors in the Coordinate Plane Let u = (x 1, y 1 ), v = (x 2, y 2 ), and let c be a scalar. 1. Scalar multiplication cu = (cx 1, cy 1 ) 2. Addition u + v = (x 1 +x 2, y 1 + y 2 ) 3. Subtraction u  v = (x 1  x 2, y 1  y 2 ) Operations on Vectors in the Coordinate Plane

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 x y u + v x y Examples: Given vectors u = (4, 2) and v = (2, 5) -2u = -2(4, 2) = (-8, -4) u + v = (4, 2) + (2, 5) = (6, 7) u  v = (4, 2)  (2, 5) = (2, -3) Examples: Operations on Vectors (4, 2) u (-8, -4) 2u2u x y (6, 7) (2, 5) (4, 2) v u (2, 5) (4, 2) v u (2, -3) u  v

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 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  =.