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Vectors and Scalars

Vectors and Scalars. A SCALAR is ANY quantity in physics that has MAGNITUDE , but NO direction. A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION.

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Vectors and Scalars

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  1. Vectors and Scalars

  2. A SCALAR is ANY quantity in physics that has MAGNITUDE, but NO direction.

  3. A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude. How are velocity and speed related? Speed is a scalar – it just has magnitude. Velocity has both magnitude and direction. Example: 20 m/s = speed 20 m/s NE = velocity

  4. A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude. Any guesses as to what displacement is? (Hint, look at the units!) Displacement is the vector quantity of distance … that is, it tells how far and in what direction two things are located.

  5. How to draw vectors The length of the vector, drawn to scale, indicates the magnitude of the vector quantity. the direction of a vector is the counterclockwise angle of rotation which that vector makes with due East or x-axis.

  6. x displacement x = 6 cm, 250 How to draw vectors – lady bug displacement NOTE 1: Displacement shows how far apart something is now compared tofrom where it started. It does not show show the path or the total distance travelled. length = magnitude 6 cm 250 above x-axis = direction NOTE 2: When drawing a vector you MUST MUSTMUST put a ‘head’ or arrow on the vector to demonstrate which direction it is pointing. L Head 250 H Tail

  7. What is the difference between a scalar and a vector? What are the parts of a vector? Quick Review

  8. What is the difference between a scalar and a vector? Scalars have magnitude only, vectors have magnitude and direction. What are the parts of a vector? Quick Review tail head

  9. Example of a vector velocity of a plane A resultant (the real one) velocity is sometimes the result of combining two or more velocities.

  10. km km km h h h 80 280 200 Adding Vectors – Plane example 1 (tailwind) A small plane is heading south at speed of 200 km/h (This is what the plane is doing relative to the air around it) To understand how far the plane is traveling relative to the ground we need to add the two vectors – the plane’s heading and the tailwind. We add vectors by moving them head to tailand finding the resultant(sum). 1. The plane encounters a tailwind of 80 km/h. km 200 h e The resulting velocity relative to the ground is 280 km/h S 80 km/ h

  11. 80 km km km h h h 120 200 Adding Vectors – Plane example 2 (headwind) A small plane is heading south at speed of 200 km/h (This is what the plane is doing relative to the air around it) 2. It’s Texas: the wind changes direction suddenly 1800. Now the plane encounters a 80 km/h headwind How do we figure out the plane’s velocity relative to the ground? Move the vectors head-to-tail and find the resultant vector. The resultant vector always goes from the tail of the first vector to the head of the second vector e The resultant velocity is only 120 km/h south..

  12. Adding or subtracting vectors in a straight line is easy, but what if the wind is coming from the side?

  13. Adding or subtracting vectors in a straight line is easy, but what if the wind is coming from the side? We need to use trigonometry.

  14. km km km km h h h h 200 200 80 80 3. The plane encounters a crosswind of 80 km/h. Will the crosswind speed up the plane, slow it down, or have no effect? How can we find out? As in the other two examples, we have to add two velocity vectors head-to-tail in order to find the resultant vector. How can we calculate the magnitude of the resultant vector? Use trigonometry! Specifically, Pythagorean theorem. RESULTANT RESULTANT VECTOR (RESULTANT VELOCITY) v = 215 km/h SE How can I find the exact angle? Use trig! Specifically, tan-1. So the plane is traveling 215 km/h at 22o E of S. F = tan-1 (80 / 200) = 22o

  15. km km km km km h h h h h 200 280 120 215 180 So why do we use vectors in physics??? Because direction matters!

  16. The order in which two or more vectors are added does noteffect result. vectors can be moved around as long as their length (magnitude) and direction are not changed. Vectors that have the same magnitude and the same direction are the same. Adding A + B + C + D + E yields the same result as adding C + B + A + D + E or D + E + A + B + C.

  17. Quick check: Are all these vectors equal or not? How do you know? Yes! They are all equal. A vector quantity is determined by its length and direction. Its position doesn’t matter. That’s why we can move vectors around to add them.

  18. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started. 54.5 m, E 30 m, W 24.5 m, E • Example: A man walks 54.5 meters east, then again 30 meters east. Calculate his displacement relative to where he started. 54.5 m, E 30 m, E 84.5 m, E • Example: A man walks 54.5 meters east, then 30 meters north. Calculate his displacement relative to where he started. 62.2 m, NE 30 m, N 54.5 m, E

  19. BUT…..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. 62.2 m, NE 30 m, N  = 290 q 54.5 m, E So the COMPLETE final answer is : 62.2 m, 290 or 62.2 m @ 290

  20. Try the following on your own • A person walks 5m N then walks 8m S. Calculate their displacement. • A ball is thrown 25 m/s E. A tailwind of 5 m/s E is blowing. Calculate the resulting velocity. • A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north

  21. Try the following on your own. • A person walks 5m N then walks 8m S. Calculate their displacement. • A ball is thrown 25 m/s E. A tailwind of 5 m/s E is blowing. Calculate the resulting velocity. 5 m 8 m 3m South 3 m 30 m 30m East 25 m 5 m

  22. A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W 15 m/s, N q The Final Answer :

  23. Example A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 12 m, W 6 m, S 20 m, N q R 35 m, E 14 m 23 m The Final Answer:

  24. IMPORTANT NOTE: The Pythagorean theorem and trig functions can only be used for right angle triangles! Later, I will teach you how to handle vectors that meet at angles other than right angles.

  25. - A 2A 3A – A A A ½ A Multiplying vector by a scalar Multiplying a vector by a scalar will ONLY CHANGE its magnitude– not direction. Opposite vectors One exception: Multiplying a vector by “-1” does not change the magnitude, but it does reverse it's direction – 3A

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