Digital Lesson Vectors

A vector is a quantity with both a magnitude and a direction. 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Vector

Two vectors, u and v, are equal if the line segments A quantity with magnitude and direction is represented by a directed line segment PQ. 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. Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude. v u Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Directed Line Segment

The component form of vector v is written A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (v1, v2). x y P Q (v1, v2) Components of v The component form of vector v is written If point P is the origin and point Q = (5, 6), then v = Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Standard Position

Definition: Component Form and Magnitude If v is a vector with initial point P = (p1 , p2) and terminal point Q = (q1 , q2), then 1. The component form of v is 2. The magnitude (or length) of v is x y P Q (p1, p2) (q1, q2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Component Form and Magnitude

Example: Component Form and Magnitude Find the component form and magnitude of vector v that has initial point (2, 2) and terminal point (–1, 4). x y -2 2 The component form of v is 3.61 units Q = (–1, 4) P = (2, 2) The magnitude of v is Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Component Form and Magnitude

Vector Addition and Subtraction Example: Given vectors find u + v and u – v . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Vector Addition and Subtraction

Scalar Multiplication Scalar multiplication is the product of a scalar, or real number, times a vector. Example: Given vector find –2u. x -4 4 -8 y u –2u The product of –2 and u gives a vector twice as long as and in the opposite direction of u. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Scalar Multiplication

Find a unit vector in the direction of v = –1, 1. 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. Unit Vectors

 is the direction angle of the vector u. 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 y x 1 –1 – 1 (x, y) y = sin  x = cos   u  is the direction angle of the vector u. Continued. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Direction Angles

The direction angle  for v is determined by 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. Direction Angles

Definition: Dot Product This yields a scalar. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Dot Product

Properties: Dot Product Dot Product Properties: Let u, v, and w be vectors and let c be a scalar. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Properties: Dot Product

Example: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Dot Product

Definition: Angle Between Two Vectors The angle between two nonzero vectors is the angle , between their standard position vectors. Origin  v - u v u Find the angle between the vectors and Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Angle Between Two Vectors

Definition: Orthogonal Vectors The vectors u and v are orthogonal if Example: x – 2 2 y 4 Because the dot product is 0, vectors u and v are orthogonal. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Orthogonal Vectors