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Vectors

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Vectors

1. 1. Vectors
2. 2. Scalars and Vectors• A scalar is a single number that represents a magnitude – E.g. distance, mass, speed, temperature, etc.• A vector is a set of numbers that describe both a magnitude and direction – E.g. velocity (the magnitude of velocity is speed), force, momentum, etc.• Notation: a vector-valued variable is differentiated from a scalar one by using bold or the following symbol:  A a 2
3. 3. Characteristics of VectorsA Vector is something that has two and only two defining characteristics:1. Magnitude: the size or quantity2. Direction: the vector is directed from one place to another. 3
4. 4. Direction• Speed vs. Velocity• Speed is a scalar, (magnitude no direction) - such as 5 feet per second.• Speed does not tell the direction the object is moving. All that we know from the speed is the magnitude of the movement.• Velocity, is a vector (both magnitude and direction) – such as 5 ft/s Eastward. It tells you the magnitude of the movement, 5 ft/s, as well as the direction which is Eastward. 4
5. 5. Example•The direction of the vector is55° North of East•The magnitude of the vectoris 2.3. 5
6. 6. Now You TryDirection: 47° North of WestMagnitude: 2 6
7. 7. Try AgainDirection: 43° East of SouthMagnitude: 3 7
8. 8. Try AgainIt is also possible to describe thisvectors direction as 47 South of East. Why? 8
9. 9. Expressing Vectors as Ordered Pairs How can we express this vector as an ordered pair? Use Trigonometry 9
10. 10. 10
11. 11. Now You TryExpress this vector as an ordered pair. 11
12. 12. Adding Vectors Add vectors A and By A B x 12
13. 13. Adding VectorsOn a graph, add vectors using the “head-to-tail” rule: y A B x Move B so that the head of A touches the tail of BNote: “moving” B does not change it. A vector is only defined by itsmagnitude and direction, not starting location. 13
14. 14. Adding VectorsThe vector starting at the tail of A and ending at the head of B is C, the sum (or resultant) of A and B. y B C = A+ B A C x 14
15. 15. Adding Vectors• Note: moving a vector does not change it. A vector is only defined by its magnitude and direction, not starting location 15
16. 16. Adding VectorsLet’s go back to our example: y 1,5 A B 7,1 x Now our vectors have values. 16
17. 17. Adding VectorsWhat is the value of our resultant? y 7,1 B 1,5 A C xGeoGebra Investigation 17