1. Vectors in the plane Vector operations

2. Vectors A vector is a quantity with both a magnitude and a direction. Vectors are used to represent velocity, force, tension, and many other quantities. x y ( u 1, u 2 )

3. A quantity with magnitude and direction is represented by a directed line segment PQ with initial point P and terminal point Q. P Q VECTORS x y ( u 1, u 2 )

4. Standard Position 4 A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point ( u 1, u 2 ). 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 )

5. Find the component form and magnitude of the vector v with initial point P = (3, 2) and terminal point Q = (1, 1). =, 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 The magnitude of v is || v || = = = 5.

6. Find the component form and magnitude of the vector v with initial point P = (4, 7) and terminal point Q = (1, 5). Try this on your own! ||v||=13

7. Vector Properties 1. u + v = v + u 3. u + 0 = u 5. c(du) = (cd)u 7. c(u+v) = cu + cv 9. ||cv|| = |c| ||v|| 2. (u + v) + w = u + (v + w) 4. u + (-u) = 0 6. (c+d)u = cu + du 8. (c-d)u = cu - du 10. 1(u) = u, 0(u) = 0

8. Warm - up Let v= and w= FIND: a. 2v b. w-v c. v+2w a. b. c. Let v= and w= FIND: a. 2v b. w-v c. v+2w

9. Unit vector A unit vector is a vector of unit length Find the unit vector in the direction of v =