1. Vectors Lesson 13.4 Pre-AP Geometry

2. Lesson Focus This lesson defines the concept of a vector. Vectors have important applications in physics, engineering, and other applied sciences.

3. Basic Terms Vector A directed quantity that has both magnitude and direction. Magnitude A scalar value having physical units. The magnitude of a mathematical object is its size or absolute value. Direction The spatial relation between something and the course along which it points or moves.

4. Vector

5. Vector AB shows the distance and direction from point A to point B. The magnitude of vector AB is the distance from A to B. The magnitude of a vector is like the absolute value of a number. It is never a negative number. The direction to B from A is found by determining the x- component and the y-component of the vector.

6. Vector Two vectors are equal if they have the same magnitude and the same direction. Two vectors are parallel if they have the same or opposite directions. Two vectors are perpendicular if they have perpendicular directions.

7. Multiplying by a Scalar Multiplying a vector by a scalar multiplies the magnitude of the vector. If the scalar multiple is negative, then the direction of the vector is reversed.

8. Vector Sum

9. Geometer’s Sketchpad Examples 01 Vector Basics 02 Vector Basics 03 Vector Basics 04 Vector Addition 07 Vector Length (Magnitude)

10. Example #1 Given: P(-3, 4) and Q(-2, -2) Sketch vector PQ, then find the magnitude and direction of vector PQ. Solution: Magnitude: Direction:

11. Example #2 Given: P(-3, 4) and Q(-2, -2) Find: and Solution:

12. Example #3 Show that (6, -3) and (-4, 2) are parallel. Solution: The slope of (6, -3) is -1/2. The slope of (-4, 2) is -1/2. The slopes are equal so the vectors are parallel.

13. Example #4 Show that (6, -3) and (2, 4) are perpendicular. Solution: The slope of (6, -3) is -1/2. The slope of (2, 4) is 2. Since -1/2  2 = -1, the vectors are perpendicular.

14. Is a vector a ray?

15. No If B is between A and C, then ray AB and ray AC are the same ray, but vector AB  vector AC. The magnitude of vector AC is greater than the magnitude of vector AB.

16. Written Exercises Problem Set 13.4A, p.541: # 2 - 26 (even)