1. Section 6.6 Vectors

2. Overview A vector is a quantity that has both magnitude and direction. In contrast, a scalar is a quantity that has magnitude but no direction.

4. Vector Representation A vector is usually represented by a directed line segment, one that has an initial point and a terminal point. Vectors are written using a boldface letter, or an arrow over a single letter:

5. Magnitude The magnitude of a vector is its length. Use the formula for the distance between points to find the length of a vector: Two vectors are equal if they have the same magnitude and the same direction.

6. Example. Given vector v with initial point P(5, -2) and terminal point Q(-3, -4): 1.Sketch v. 2.Find the magnitude of v.

7. Unit Vectors A unit vector is a vector with a magnitude of 1. Vector i is the unit vector whose initial point is at the origin and whose direction is along the positive x-axis. Vector j is the unit vector whose initial point is at the origin and whose direction is along the positive y-axis.

9. More… Vectors in the rectangular coordinate system can be represented in terms of i and j: If vector v has initial point at the origin and terminal point (a,b), then a is the horizontal component and b is the vertical component, and

10. More… If the initial point of v is not at the origin, then

11. Examples Let v be the vector from initial point P(-3, -5) to terminal point Q(3, 4). 1.Sketch the graph. 2.Find the magnitude of v. 3.Write v in terms of i and j.

12. Vector Arithmetic in Terms of i and j Ifandand k is a real number then:

13. Examples Let u = 2i – 7j and v = -4i + 8j. Find each of the following vectors, written in terms of i and j. 1.u – v 2.7u + 5v 3.The magnitude of v – u

14. Unit Vectors re-visited For any nonzero vector v, the vector is the unit vector that has the same direction as v.

15. Example Find the unit vector that has the same direction as the vector v = 6i + 8j

16. Writing a vector in terms of its magnitude and direction Example: if vector v has a magnitude ||v|| = 32 and a direction θ = 225°, write v in terms of i and j.

17. Resultant Forces When two vectors are acting simultaneously on an object, the resultant force can be found by: 1.Writing each vector in terms of i and j, then adding the vectors together (parallelogram method). 2.Drawing the vectors from “tip to tail”, then using the Law of Sines and/or the Law of Cosines (tip to tail method) to find the magnitude and direction of the resultant force.

18. Some Pictures

19. Examples The magnitude and direction of two forces acting on an object are 110 pounds, S61°E, and 120 pounds, N54°E, respectively. Find the magnitude and direction of the resultant force. MLP, Problem 15.