Chapter 3 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 A B R = A + B

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 2D 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 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) (c) - 2.8

Answers to the even-numbered problems Chapter 3: Problem 38: (a) 57° (b) 2.2 m (c) m (d) 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^