8.2.3 Vectors Key Objectives • Find the magnitude and direction of a vector. • Use vectors and vector addition to solve real-world problems. Key Terms • A vector is a quantity that has both length and direction. • The component form 〈 x , y 〉 of a vector lists the horizontal and vertical change from the initial point to the terminal point. • The magnitude of a vector is its length. • The direction of a vector is the angle that it makes with a horizontal line. The angle is measured counter- clockwise from the positive x -axis. • Two vectors are equal vectors if they have the same magnitude and the same direction. • Two vectors are parallel vectors if they have the same direction or if they have opposite directions. • The resultant vector is the vector that represents the sum of two given vectors. Example 1 Writing Vectors in Component Form A vector in the coordinate plane is written in component form in this example.
The vector direction is from the initial point A to the terminal point B . The horizontal change in the vector is 4 units. The vertical change in the vector is 6 units. The component form of vector AB is 〈 4, 6 〉 .
A vector in the coordinate plane is written in component form in this example. The coordinates of the endpoints of the vector are given. The initial point of the vector is P and the terminal point of the vector is Q . The components are determined by subtracting the initial-point x coordinate from the terminal-point x coordinate, and the initial point y coordinate from the terminal point y coordinate. The component form of vector PQ is 〈− 10, − 2 〉 .
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