It’s time for Chapter 6… Section 6.1a Vectors in the Plane

Scalar – a single real number, has magnitude Vector – has magnitude and direction  Examples??? Vectors are denoted by lowercase boldface letters (e.g., u, v, w), and can be defined by directed line segments

Our first example: Consider directed line segment PQ, with P(– 4, 2) and Q(1, 1): Notation! P Q y x We can find the length, or magnitude of the segment, |PQ|, using the distance formula: |PQ| = 26 Initial Point Terminal Point

Another example: Let u be the vector represented by the directed line segment from R(– 4, 2) to S(–1, 6) and v the vector represented by the directed line segment from O(0, 0) to P(3, 4). Prove that u = v. How ‘bout a graph? Lengths? Slopes?  The two vectors are equivalent  a vector is defined by its direction and magnitude, not by its location direction and magnitude, not by its location

Definition: Component Form of a Vector If v is a vector in the plane equal to the vector with initial point (0, 0) and terminal point (v, v ), then the component form of v is: 12 v = v, v 12 Components This vector is called the position vector of the point (v, v ) 12 The vector 0, 0 with length 0 and no direction is called the zero vector, and is denoted 0

We need some more practice examples… Find the component form and magnitude of the vector v = PQ, where P(–3, 4) and Q(–5, 2). Let’s see a graph: v = –2, –2 |v|= 2 2

We need some more practice examples… Find the component form and magnitude of the vector u = FG, where G(1, –2) and F(7, 1). Let’s see a graph: u = – 6, –3 |u|= 3 5

Vector Operations Sum of vectors: u + v = Product of a scalar and a vector: ku = Would this work for subtraction as well??? Would this work for a product of two vectors???

Let’s see these graphically y x (4, 2) (– 6, 6) u v What is u + v? u + v (– 2, 8) The parallelogram law…

Let’s see these graphically y x (2, 3) u If k = 3, what is ku? (6, 9) 3u3u

Practice Problems Let u = –2, 8 and v = 3, – 5. Find the component form of: u + 2v 4, –2

Unit Vectors Unit vector – a vector u with length |u| = 1 Unit vector in the direction of v: u = = v |v||v| 1 |v||v| v

Unit Vectors Standard unit vectors:i = 1, 0 and j = 0, 1 Any vector w can be written as an expression using the standard unit vectors: w = a, b = a, 0 + 0, b = a 1, 0 + b 0, 1 = ai + bj Horizontal and vertical components of w In a graph?

How ‘bout some examples? Find a unit vector in the direction of u = 7, 1, and verify that it has length 1. Unit Vector: Magnitude:

Whiteboard Problem… Let P = (–2,2), Q = (3,4), R = (–2,5), and S = (2,–8). Find the component form and magnitude of

Whiteboard Problems Let u = –2, 8 and v = 3, – 5. Find the component form of: 3u – 4v –18, 44

Find the unit vector in the direction of v = 4, –2. Write your answer in both component form and as a combination of the standard unit vectors. Unit Vector: Component Form:Standard Unit Vectors: Whiteboard Problem…