Chapter 3: Vectors

1. Chapter 3: Vectors Things to ask the class: Have you all checked your WebCT account? Do the Quiz, participate in the discussion and learn from the applets! • In this chapter we will learn about vectors, (properties, addition, components of vectors and multiplying) Dr. Mohamed S. Kariapper - KFUPM

2. Vectors: Magnitude and direction Scalars: Only Magnitude A scalar quantity has a single value with an appropriate unit and has no direction. Examples for each: Vectors: Scalars: Motion of a particle from A to B along an arbitrary path (dotted line). Displacement is a vector Dr. Mohamed S. Kariapper - KFUPM

3. Vectors: • Represented by arrows (example displacement). • Tip points away from the starting point. • Length of the arrow represents the magnitude • In text: a vector is often represented in bold face (A) or by an arrow over the letter. • In text: Magnitude is written as A or This four vectors are equal because they have the same magnitude and same length Dr. Mohamed S. Kariapper - KFUPM

4. Adding vectors: Graphical method (triangle method): Draw vector A. Draw vector B starting at the tip of vector A. The resultant vector R = A + B is drawn from the tail of A to the tip of B. Dr. Mohamed S. Kariapper - KFUPM

5. Adding several vectors together. Resultant vector R=A+B+C+D is drawn from the tail of the first vector to the tip of the last vector. Dr. Mohamed S. Kariapper - KFUPM

6. Commutative Law of vector addition A + B = B + A (Parallelogram rule of addition) Dr. Mohamed S. Kariapper - KFUPM

7. Associative Law of vector addition A+(B+C) = (A+B)+C The order in which vectors are added together does not matter. Dr. Mohamed S. Kariapper - KFUPM

8. Negative of a vector. The vectors A and –A have the same magnitude but opposite directions. A + (-A) = 0 A -A Dr. Mohamed S. Kariapper - KFUPM

9. Subtracting vectors: A - B = A + (-B) Dr. Mohamed S. Kariapper - KFUPM

10. Multiplying a vector by a scalar The product mA is a vector that has the same direction as A and magnitude mA. The product –mA is a vector that has the opposite direction of A and magnitude mA. Examples: 5A; -1/3A Dr. Mohamed S. Kariapper - KFUPM

11. Components of a vector The x- and y-components of a vector: The magnitude of a vector: The angle q between vector and x-axis: Dr. Mohamed S. Kariapper - KFUPM

12. The signs of the components Ax and Ay depend on the angle q and they can be positive or negative. (Examples) Dr. Mohamed S. Kariapper - KFUPM

13. Unit vectors • A unit vector is a dimensionless vector having a magnitude 1. • Unit vectors are used to indicate a direction. • i, j, k represent unit vectors along the x-, y- and z- direction • i, j, k form a right-handed coordinate system Dr. Mohamed S. Kariapper - KFUPM

14. The unit vector notation for the vector A is: OR in even better shorthand notation: Dr. Mohamed S. Kariapper - KFUPM

15. Adding Vectors by Components We want to calculate: R = A + B From diagram: R = (Axi + Ayj) + (Bxi + Byj) R = (Ax + Bx)i + (Ay + By)j Rx = Ax+ Bx Ry = Ay+ By The components of R: Dr. Mohamed S. Kariapper - KFUPM

16. Adding Vectors by Components The magnitude of a R: The angle q between vector R and x-axis: Dr. Mohamed S. Kariapper - KFUPM

17. Checkpoint 3 (page 45) What are the signs of x components of d1 and d2? What are the sign of the y components of d1 and d2? What are the signs of the x and y components of d1+d2 Dr. Mohamed S. Kariapper - KFUPM

18. Sample Problem 3-4 Dr. Mohamed S. Kariapper - KFUPM

19. (a) The vectors and its components (b) The same vector, with the axes of the coordinate system rotated through an angle Vectors and the Laws of Physics: This section will be left as a reading assignment for the student. Dr. Mohamed S. Kariapper - KFUPM

20. Multiplying a vector by another vector: Two types: Dr. Mohamed S. Kariapper - KFUPM

21. Dr. Mohamed S. Kariapper - KFUPM