1. Adding Vectors Graphically CCHS Physics

2. Vectors and Scalars Scalar has only magnitude Vector has both magnitude and direction –Arrows are used to represent vectors –The direction of the arrow gives the direction of the vector –The length of a vector arrow is proportional to the magnitude of the vector

3. Vector Properties Notation –When vector is handwritten, often shown with arrow or other designation –In book, usually bold face type, ex: A –Magnitude of A represented by italic, ex: A Equality of Vectors –Two vectors, A and B, are defined as equal if they have the same magnitude and direction

4. Vector Properties Cont. Vector Addition (graphically) –All the vectors must have the same units –Tip-to-Tail Method of Addition Draw vector A to scale (ie 1 cm = 1 m) Then draw vector B to the same scale with the tail of B starting at the tip of A Resultant vector R is given by R = A + B

5. Vector Properties Movie

6. Vector Properties Cont. –Parallelogram Method of Addition The tails of vectors A and B are joined, and the resultant vector, R, is the diagonal of the parallelogram formed with A and B as its sides. –Note A + B = B + A –To add more than two vectors, just continue adding tail to tip.

7. Vector Properties –Negative of a Vector When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction is reversed

8. Vector Properties Cont. Vector Subtraction (graphically) –Carried out exactly like vector addition, except that one of the vectors is multiplied by a scalar factor of -1 – A - B = A + (- B)

9. Subtracting Vectors

10. Vector Properties Cont. Multiplication and Division of Scalar by Vectors –Multiplication or division of vector by a scalar yields a vector –If the given vector B is multiplied by the scalar 4, the result, written 4B, is a vector with a magnitude four times the original vector B, pointing in the same direction as B. B4B4B

11. Multiplication by Scalar

12. Adding Vectors Graphically

13. Displacement Hike 4 km East 2 km NE 3 km @ 120° 5 km @ 210° R = 1.6 km @ 105° START FINISH Scale: 1” = 1km

14. ADDING VECTORS MATHEMATICALLY CCHS PHYSICS

15. Components Components: projections of a vector along axes of rectangular coordinate system –Can resolve vectors into components

16. Trigonometry Review h = length of hypotenuse of right triangle h o = length of side opposite the angle  h a = length of side adjacent to the angle 

17. Trig Review Cont. Inverse Trigonometric Functions Pythagorean Theorem

18. Finding Vector Components

19. Adding Vectors Mathematically Select coordinate system Resolve all the vectors into components Add all the x-components Add all the y-components –The sum of the x and y components gives you the components of the resultant Find the magnitude of the resultant via the Pythagorean Theorem Find the angle with a suitable trig function

20. Adding Mathematically

21. Displacement Hike Revisited Given the following vectors, mathematically determine the resultant: 4 km East 2 km NE 3 km @ 120° 5 km @ 210°

22. Displacement Hike Revisited ActionX ComponentY Component 4 km E x = 4 kmy = 0 km 2 km NE x = 2cos45 x = 1.4 km y = 2sin45 y = 1.4 km 3 km @ 120° x = -3cos60 = -1.5 km y = 3sin60 = 2.6 km 45° 2 km 4 km 120° 3 km 60°

23. Displacement Hike Revisited ActionX ComponentY Component 5 km @ 210° x = -5cos30 x = -4.3 km y = -5sin30 y = -2.5 km RESULTANT  x = 4+1.4-1.5-4.3 =-0.4 km  y = 0+1.4+2.6-2.5 =1.5 km Magnitude: Direction: 210° 5 km 30° 105° R = 1.6 km 75° R x = -.4 R y = 1.5

24. Resultant and Equilibrant Resultant: the single vector (usually with regards to force) that is equal to two or more other vectors Equilibrant: the single vector (usually with regards to force) that will balance two or more vectors –Equal in magnitude opposite in direction to the resultant F1F1 F2F2 Resultant Equilibrant

25. 45° 2 km 210° 5 km 30° F1F1 F2F2 Resultant Equilibrant 120° 3 km 60° 105° R = 1.6 km 75° R x = -.4 R y = 1.5