Exploring Vectors: Magnitude, Direction & Addition

1. Warm Up • Find AB when A has coordinates (2,5) and B(-3, 2) • Find the slope of the two points: (-2, 5) and (3, -1)

2. 9.1 Vectors Geometry Mr. Jimenez

3. Objectives: • Find the magnitude and the direction of the vector • Add vectors

4. Homework • Page 576 #16-28 even • What do vectors do? • If I start with (0, 2) and move the point to (4, 6); the vector would be <4, 4> because the x moved 4 units and the y moved 4 units.

5. Vectors track movement • Vectors track the way the point moves by the way it moves left or right (the x-value) and the way it moves up or down (the y-value. • Vectors detect direction as said before as well as magnitude or the force of an object.

6. You begin with an initial point to a terminal point given in terms of points, usually P and Q. You graph it as you would a ray. Initial point is P(0, 0). Terminal point is Q(-6, 3). Finding the magnitude of a vector Q(-6, 3) P(0, 0)

7. Here you write the following Component Form =‹x2 – x1, y2 – y1› <-6 – 0, 3 – 0> <-6, 3> is the component form. Next use the distance formula to find the magnitude. The direction is given by this. |PQ| = √(-6 – 0)2 + (3 – 0)2 = √62 + 32 = √36 + 9 = √45 ≈ 6.7 Write the component form Q(-6, 3) P(0, 0)

8. Initial point is P(0, 2). Terminal point is Q(5, 4). Reminder that Q is the second point. P is the initial point. Graph the ray starting at P and going through Q as to the right. Then you can start looking for component form and magnitude. Graph Initial/Terminal points

9. Here you write the following Component Form =‹x2 – x1, y2 – y1› <5 – 0, 4 – 2> <5, 2> is the component form. Next use the distance formula to find the magnitude. |PQ| = √(5 – 0)2 + (4 – 2)2 = √52 + 22 = √25 + 4 = √29 ≈ 5.4 Write the component form

10. Initial point is P(3, 4). Terminal point is Q(-2, -1). Reminder that Q is the second point. P is the initial point. Graph the ray starting at P and going through Q as to the right. Then you can start looking for component form and magnitude. Graph Initial/Terminal points

11. Here you write the following Component Form =‹x2 – x1, y2 – y1› <-2 – 3, -1 – 4> <-5, -5> is the component form. Next use the distance formula to find the magnitude. |PQ| = √-2 – 3)2 + (-1– 4)2 = √(-5)2 + (-5)2 = √25 + 25 = √50 ≈ 7.1 Write the component form

12. Adding Vectors • Two vectors can be added to form a new vector. To add u and v geometrically, place the initial point of v on the terminal point of u, (or place the initial point of u on the terminal point of v). The sum is the vector that joins the initial point of the first vector and the terminal point of the second vector. It is called the parallelogram rule because the sum vector is the diagonal of a parallelogram. You can also add vectors algebraically.

13. What does this mean? • Adding vectors: Sum of two vectors The sum of u = <a1,b1> and v = <a2, b2> is u + v = <a1 + a2, b1 + b2> In other words: add your x’s to get the coordinate of the first, and add your y’s to get the coordinate of the second.

14. Example: • Let u = <3, 5> and v = <-6, 1> • To find the sum vector u + v, add the x’s and add the y’s of u and v. u + v = <3 + (-6), 5 + (-1)> = <-3, 4>

15. Try it: • Let u = <4, -1> and v = <2, 1> • To find the sum vector u + v, add the x’s and add the y’s of u and v. u + v = <4 + (2), (-1) + (1)> = <6, 0>