fisica1-youn-e-freedman-exercicios-resolvidos

1.48: a) ˆ ˆ ˆ 2 2 2
i + j + k = 1 + 1 + 1 = 3 ≠ 1 so it is not a unit vector
b)
A r
=
A + A + A
2
x
2
y
2
z
If any component is greater than + 1 or less than –1, A r
≥ 1, so it cannot be a unit
vector. A r can have negative components since the minus sign goes away when the
component is squared.
c)
a
2
2 2
( 3.0) + a ( 4.0)
a
2
2
= 1
= 1
25 = 1
A r 20
r
1.49: a) Let A = A iˆ
ˆ
x
+ Ay
j,
r r
A + B = A iˆ
x
+ Bx
+ A
r r
B + A = B + A iˆ
+ B
a = ±
1
5.0
( ) ( + B )
( ) ( + A )j ˆ
x
x
Scalar addition is commutative, so
r r
A ⋅ B = Ax
Bx
+ AyBy
r r
B ⋅ A = B A + B A
x
x
y
y
y
y
y
y
= ± 0.
r
B = B iˆ
+ B
ˆj
x
y
ˆ. j
r r r r
A + B = B + A
r r r r
Scalar multiplication is commutative, so A ⋅ B = B ⋅ A
b) A
r × B
r
= ( A ) iˆ
( ) ˆj
( )kˆ
yBz
− Az
By
+ Az
Bx
− Ax
Bz
+ Ax
By
− Ay
Bx
B
r
× A
r
= ( B A − B A ) iˆ
+ ( B A − B A ) ˆj
+ ( B A − B A ) kˆ
y
z
z
y
z
x
Comparison of each component in each vector product shows that one is the
negative of the other.
x
z
x
y
y
x