1. Chapter 3
Vectors

2. Vectors
Vectors – physical quantities having both magnitude and direction
Vectors are labeled either a or
Vector magnitude is labeled either |a| or a
Two (or more) vectors having the same magnitude and direction are identical

3. Vector sum (resultant vector)
Not the same as algebraic sum
Triangle method of finding the resultant:
Draw the vectors “head-to-tail”
The resultant is drawn from the tail of A to the head of B
R = A + B B A

4. Addition of more than two vectors
When you have many vectors, just keep repeating the process until all are included
The resultant is still drawn from the tail of the first vector to the head of the last vector

5. Commutative law of vector addition
A + B = B + A

6. Associative law of vector addition
(A + B) + C = A + (B + C)

7. Negative vectors
Vector (- b) has the same magnitude as b but opposite direction

8. Vector subtraction
Special case of vector addition: A - B = A + (- B)

9. Multiplying a vector by a scalar
The result of the multiplication is a vector
c A = B
Vector magnitude of the product is multiplied by the scalar
|c| |A| = |B|
If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector

10. Vector components
Component of a vector is the projection of the vector on an axis
To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector

11. Vector components

12. Unit vectors Unit vector: Has a magnitude of 1 (unity)
Lacks both dimension and unit
Specifies a direction
Unit vectors in a right-handed coordinate system

13. Adding vectors by components
In 2D case:

14. Chapter 3
Problem 33
Vector B has z, y, and z components of 4.00, 6.00, and 3.00 units, respectively. Calculate the magnitude of B and the angles B makes with the coordinate axes.

15. Answers to the even-numbered problems
Chapter 3
Problem 4:
y = 1.15; r = 2.31

16. Answers to the even-numbered problems
Chapter 3
Problem 6:
310 km at 57° S of W

17. Answers to the even-numbered problems
Chapter 3
Problem 16:
1.31 km north; 2.81 km east

18. Answers to the even-numbered problems
Chapter 3
Problem 20:
m i^ m j^

19. Answers to the even-numbered problems
Chapter 3
Problem 24:
(b) 5.00 i^ j^, at 38.7°;
– 1.00 i^ j^, 8.06 at 97.2°

20. Answers to the even-numbered problems
Chapter 3
Problem 30:
C = 7.30 cm i^ cm j^

21. Answers to the even-numbered problems
Chapter 3
Problem 52:
(a) 2.00, 1.00, 3.00
(b) 3.74
(c) θx = 57.7°, θy = 74.5°, θz = 36.7°