Chapter 3 Vectors. Vectors – physical quantities having both magnitude and direction Vectors are...
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Transcript of Chapter 3 Vectors. Vectors – physical quantities having both magnitude and direction Vectors are...
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^