10.1.1 Introduction to Vectors

1. 10.1.1 Introduction to Vectors Today you will use geometry to define and perform operations using vectors. You will also write vectors in component form.

2. An Introduction to Vectors • When we describe physical phenomenon such as wind or a current in a river, we look at the direction of the force and its strength. • These properties can be described by using vectors. • Vectors are used to describe a wide variety of real world forms such as wind, velocity, and force because they have both magnitude and direction.

3. Simon Says Vector Game • Defining our space: North South East West • One partner will “act out” the problem while the other partner will record the motions on the graph paper. • Halfway through you will change roles with your partner.

4. Simon Says Vector Game #1 • Go three steps east. • Go three steps south. • Go one step west. • Go two steps south. • Go four steps west. • Go two steps north.

5. Simon Says Vector Game #2 • Go two steps north. • Go three steps west. • Go one step east. • Go three steps south. • Go four steps east. • Go one step south. • Go four steps west. • Go one step south.

7. Shall We Dance? • In the Simon Says activity, if the steps were followed correctly, did each person make the same movements? • Was everyone in the same location? • With your partner, complete the Sketch column of the Vector Line Dance Activity. Start each step at the ending point of the previous step.

9. Vector Line Dance • The vector represents the movement in the horizontal and vertical directions (x and y). • The angle is the measurement from the positive x-axis (this is called the standard angle). • The length of the vector is called the magnitude.

10. Vectors The arrows you have been drawing are called vectors. 4 steps 4 steps • Do these two vectors represent the same instruction? • Are the starting points the same? • Two vectors that represent the same instruction are called equivalent.

11. Magnitude & Direction • Vectors have both a length called magnitude and a direction. Looking at the instructions in the activity; find vectors that are equivalent in magnitude, but go in opposite directions. Step 1 and Step 5 Step 4 and Step 7

12. Angle and Magnitude • Find the Angle and Magnitude for each step on the resource page. • To find the angle, think of the starting point for each vector as the origin and figure out the angle to the vector from the positive horizontal axis. • The Magnitude is the length of the vector. Draw a right triangle and use the Pythagorean theorem if necessary.

14. Vector Notations

15. Component Form

16. Component Form • With your partner, find the component form of each vector on the resource page.

18. Equivalent Vectors • In the graphs below, vectors p and q are called equivalent vectors. Equivalent vectors are vectors that have the same magnitude and direction. • Name two other pairs of equivalent vectors • Draw a vector equivalent to x. Are everyone’s answers equivalent?

19. Adding Vectors Geometrically • Copy vectors r and m on a sheet of graph paper. • Step 1: Draw a vector equivalent to m whose initial point coincides with the endpoint of r. • Step 2: Draw a vector from the initial point of r to the endpoint of m. • Step 3: Label this vector r + m. We call this new vector the sum, or resultant, of r and m. This is called the head to tail method for adding vectors.

20. Adding Vectors Geometrically • Draw another vector labeled a which is equivalent to r + m. • Copy vector p onto your graph paper. Then draw r + p. Label the resultant vector b. • Find the component forms for vectors r, p, and b. Using your observations, make a conjecture about adding vectors in component form. Test your conjecture by using your results from adding r and m.

21. More Adding Vectors In the diagram above, vector v is added to vector u (not shown) to get the resultant vector w. What is the component form of u?

22. Assignment HW 10.1.1 10-6 to 10-14 (pg. 476)