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 10

Chapter 3: Problem 20

Scalar product of two vectors • The result of the scalar (dot) multiplication of two vectors is a scalar • Scalar products of unit vectors

Scalar product of two vectors • The result of the scalar (dot) multiplication of two vectors is a scalar • Scalar product via unit vectors

Vector product of two vectors • The result of the vector (cross) multiplication of two vectors is a vector • The magnitude of this vector is • Angle φ is the smaller of the two angles between and

Vector product of two vectors • Vector is perpendicular to the plane that contains vectors and its direction is determined by the right-hand rule • Because of the right-hand rule, the order of multiplication is important (commutative law does not apply) • For unit vectors

Vector product in unit vector notation

Answers to the even-numbered problems Chapter 3: Problem 12: (a) 12 (b) - 5. 8 (c) - 2. 8

Answers to the even-numbered problems Chapter 3: Problem 38: (a) 57° (b) 2. 2 m (c) - 4. 5 m (d) - 2. 2 m (e) 4. 5 m

Answers to the even-numbered problems Chapter 3: Problem 58: (a) 8 i^ + 16 j^ (b) 2 i^ + 4 j^