Chapter 3 Vectors Vectors Vectors physical quantities having

Chapter 3 Vectors

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

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

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

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

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

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

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

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

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

Vector components

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

Adding vectors by components In 2 D case:

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.

Answers to the even-numbered problems Chapter 3 Problem 4: y = 1. 15; r = 2. 31

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

Answers to the even-numbered problems Chapter 3 Problem 16: 1. 31 km north; 2. 81 km east

Answers to the even-numbered problems Chapter 3 Problem 20: - 25. 0 m i^ + 43. 3 m j^

Answers to the even-numbered problems Chapter 3 Problem 24: (b) 5. 00 i^ + 4. 00 j^, 6. 40 at 38. 7°; – 1. 00 i^ + 8. 00 j^, 8. 06 at 97. 2°

Answers to the even-numbered problems Chapter 3 Problem 30: C = 7. 30 cm i^ - 7. 20 cm j^

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°