1. 3.1 & 3.2 Vectors & Scalars

2. Biblical Reference The Lord will grant that the enemies who rise up against you will be defeated before you. They will come at you from one direction but flee from you in seven. Deuteronomy 28:7

3. Scalar Scalar ExampleMagnitude Speed20 m/s Distance10 m Age15 years Heat1000 calories Any quantity in physics that has Magnitude, but No Direction Magnitude: A numerical value with units.

4. Vector Magnitude & Direction Velocity20 m/s, N Acceleration10 m/s/s, E Force5 N, West Click Here for a better explanation of vectors. Any quantity in physics that has Both Magnitude and Direction Typically illustrated with an arrow above the symbol to convey magnitude & direction

5. Vector Addition – If 2 similar vectors point in the same direction, add them. Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? Applications of Vectors 54.5 m E30 m E + 84.5 m E Notice that the Size of the arrow conveys Magnitude and the way it was drawn conveys Direction. Resultant – A vector that represents the sum or 2 or more vectors

6. Vector Subtraction - If 2 vectors are going in opposite directions, you Subtract. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? Applications of Vectors 54.5 m E 30 m W - 24.5 m E

7. The addition of another axis helps to describe motion. Two Dimensional Coordinate Systems

8. Non-Collinear Vectors When 2 vectors are perpendicular, use the Pythagorean Theorem. 95 km E 55 km N Start Finish Example: A man walks 95 km E then 55 km N. Calculate his Resultant Displacement. The hypotenuse is called the Resultant. The Legs of the triangle are called the Components. Horizontal Component Vertical Component

9. You should also include a Direction for all vectors. BUT……what about the direction? Note: When drawing a right triangle that conveys some type of motion, you MUST draw your components Head-to-Toe. N S E W N of E E of N S of W W of S N of W W of N S of E E of S N of E

10. BUT...what about the VALUE of the angle? Just putting North of East on the answer is Not specific enough for the direction. You Must find the Value of the angle. N of E 55 km N 95 km E To find the value of the angle, use a Trig function called Tangent. q 109.8 km The Complete final answer is : 109.8 km, 30  North of East

11. Example – Find the Resultant While following the directions on a treasure map, a pirate walks 45.0 m north and then turns and walks 7.50 m east. What is the single straight line displacement? 45 m N 7.5 m E East of North or 80.54  North of East

12. What if you are missing a component? Suppose a person walked 65 m, 25  East of North. What were his horizontal and vertical components? 65 m 25 H.C. = ? V.C = ? The goal: Always make a Right Triangle! To solve for components, use the trig functions sine and cosine.

13. Example – Find components A helicopter flies at 95 km/hr at an angle of 35  to the ground. Find the components of its motion. 95 m 35 H.C. = ? V.C = ? ground

14. Example A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s W. Calculate the boat's resultant velocity with respect to due north. 15 m/s, N 8.0 m/s, W RvRv  The Final Answer: 17 m/s, @ 28.1  West of North

15. Example A plane moves with a velocity of 63.5 m/s at 32  South of East. Calculate the plane's horizontal and vertical velocity components. 63.5 m/s 32 H.C. =? V.C. = ?

16. Example A storm system moves 5000 km due east, then shifts course at 40  N of E for 1500 km. Calculate the storm's resultant displacement. 5000 km, E 40 1500 km H.C. V.C. 5000 km + 1149.1 km = 6149.1 km 6149.1 km 964.2 km R  Final Answer: 6224.2 km @ 8.9  N of E

17. Adding Non-Perpendicular Vectors Many objects move in one direction and then turn at an angle before continuing their motion. Break the displacement vectors into x and y components.

18. Example – Non-Perpendicular Vectors A bus travels 301 m 23° above the x-axis. Then it turns and travels 235 m 12° above the x-axis. What is the displacement of the bus? =+ A B

19. Example – Non-Perpendicular Vectors A 23 301 m

20. Example – Non-Perpendicular Vectors B 12 235 m

21. Example – Non-Perpendicular Vectors

22. What if we subtract the second vector from the first in the previous problem? A = 301 m 23° above the x-axis B = 235 m 12° above the x-axis A – B = A + (-B) Example – Subtracting Two Vectors

23. A = 301 m 23° above the x-axis B = 235 m 12° above the x-axis

24. Last Example A man walks 26 m East and then walks 35 m 27  North of East. What is his displacement? 26 m, E 27 35 m H.C. V.C. 26 m + 31.2 m = 57.2 m 57.2 m 15.9 m R q Final Answer: 59.4 m @ 15.5  N of E