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

a

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

x

y

yx a

aaaa tan

22

inaaaa yx s cos

Unit vectors

• Unit vector:A) Has a magnitude of 1 (unity)B) Lacks both dimension and unitC) Specifies a direction

• Unit vectors in a right-handed coordinate system

Adding vectors by components

In 2D case:

jbibb

jaiaa

yx

yx

ˆˆ

ˆˆ

bar

yyy

xxx

bar

bar

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

cosabba

ii ˆˆ 1ˆˆ jj

ji ˆˆ

1ˆˆ kk

0ˆˆ ki 0ˆˆ kj

0cos11 1

90cos11 0

Scalar product of two vectors

• The result of the scalar (dot) multiplication of two vectors is a scalar

• Scalar product via unit vectors

cosabba

)ˆˆˆ)(ˆˆˆ( kbjbibkajaiaba zyxzyx

zzyyxx babababa

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

cba

sinabc

b

a

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

)( baab

c

b

a

ii ˆˆ 0 kkjj ˆˆˆˆ

ji ˆˆ k̂ ikj ˆˆˆ jik ˆˆˆ

Vector product in unit vector notation

)ˆˆˆ()ˆˆˆ( kbjbibkajaiaba zyxzyx

ibia xxˆˆ

jbia yxˆˆ

kabbajabba

iabbaba

yxyxxzxz

zyzy

ˆ)(ˆ)(

ˆ)(

)ˆˆ( iiba xx 0

)ˆˆ( jiba yx kba yxˆ

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^