1. Vectors Angle Reference direction

2. Vector A is identical to Vector B, just transported (moved on a graph keeping the same orientation and length). Vector A Vector B

3. Cartesian CCW = + Compass CW = + 1 2 3 4

5. How to show magnitude of vectors - mathematically and graphically

6. Adding two vectors graphically A + B = R Head to tail method

7. Showing A + B = B + A

8. Showing A - B ≠ B - A Tail to tail method

9. Showing A - B = A + (- B)

10. Breaking vectors down in component parts

11. V = V x + V y + V z

12. Step 1: Break down vectors to be added into there Vx and Vy components (for three dimension x, y and z components) Step 2: Sum the Vx and then Vy components. Step 3 use the Pythagorean theorem to solve for the magnitude resultant vector Step 4: Use SOH-COA-TOA to find the vector angel from the x axis Example: Add vector A =10 that points to 030º (Cart) with a vector B = 20 that points to 060º (Cart) A B Step 1: Break vectors into components A = Ax + Ay Ax = Cos 30º (10) = 8.67 Ay = Sin 30º (10) = 5 B = Bx + By Bx = Cos 60º (20) = 10 By = Sin 60º (20) = 17.3 Adding Vectors mathematically

13. Step 2: Solve for Vx an Vy Vx = Rx = Ax + Bx = 8.67 + 10 = 18.67 Vy = Ry = Ay + By = 5 + 17.3 = 22.3 Step 3: Solve for R (magnitude) |R| 2 = Vx 2 + Vy 2 |R| 2 = 18.67 2 + 22.3 2 |R| 2 = 348.57 + 497.29 = 845.86 |R| = (845.86) 1/2 |R| = 29.1 Step 4: Solve for an angle Tan (Vector Angle - from x axis) = 22.3/18.67 = 1.194 Tan -1 (1.194) = 50.1º

14. A B Graphical Check B A 10

15. A B A A + B B A Ay = 5 Ax = 8.67 By = 17.3 Bx = 10 B Ry = 17.3 + 5 = 22.3 Rx = 8.67 + 10 = 18.67 R = A + B = 29.1 Angle = Tan -1 22.3/18.67 = 50.1º