Chapter 2
Vector Algebra
Review
Dr. Ray Kwok
SJSU
Vector Algebra - Dr. Ray Kwok
Vector products
(scalar) (scalar) = scalar
scalar, (a)(b) = ab
e.g. 2(4 kg) = 8 kg
(scalar) (vector) = vector,
e.g. 5(2 xˆ + 4 yˆ ) = 10 xˆ + 20 yˆ
r
r
k A = kA
()
(vector) times (vector) = ?
r r
can be scalar (scalar product, or dot product) A ⋅ B
or
r r
vector (vector product, or cross product) A × B
Can you add a vector to a scalar?
Vector Algebra - Dr. Ray Kwok
The scalar product (dot product)
r r
A ⋅ B = AB cos θ
x̂ ⋅ x̂ = 1 = ŷ ⋅ ŷ = ẑ ⋅ ẑ
x̂ ⋅ ŷ = 0 = x̂ ⋅ ẑ = ŷ ⋅ ẑ
r r
A ⋅ B = (A x x̂ + A y ŷ + A z ẑ ) ⋅ (Bx x̂ + B y ŷ + Bz ẑ )
r r
A ⋅ B = A x B x + A y B y + A z Bz
e.g.
r
A = 2x̂ − 2 ŷ
r
B = 3x̂ + ŷ
What is A · B ?
e.g.
What’s the
angle between
these 2 arrows?
Vector Algebra - Dr. Ray Kwok
Interpretation - projection
Vector Algebra - Dr. Ray Kwok
Example - Work
r r
W = F ⋅ ∆x
(Projection)
So, is the uplifting force doing anything at all??
Vector Algebra - Dr. Ray Kwok
The vector product (cross product)
r r
A × B = AB sin θ
x̂ × x̂ = 0 = ŷ × ŷ = ẑ × ẑ
x̂ × ŷ = ẑ
right-hand coordinate
ŷ × ẑ = x̂
cyclic permutation
ẑ × x̂ = ŷ
ŷ × x̂ = −ẑ
ẑ × ŷ = − x̂
x̂ × ẑ = − ŷ
e.g.
r
A = 2x̂ − 2 ŷ
r
B = 3x̂ + ŷ
What is A x B ?
x̂
r r
A × B ≡ Ax
ŷ
ẑ
Ay
Az
Bx
By
Bz
e.g.??
What’s the
angle between
these 2 arrows?
Vector Algebra - Dr. Ray Kwok
Example – calculate torque
r r
τ = r ×F
r
4N
x
o
40o
1.5 m
choose “+” = counter-clockwise
Στo = (1.5)(4)sin(40o)
= 3.86 N-m (counter-clockwise)
Vector Algebra - Dr. Ray Kwok
Example – perpendicular F
4N
40o
x
1.5 m
o
4 sin(40o)
1.5 m
x
o
Στo = (1.5)[4 sin(40o)]
= 3.86 N-m (counter-clockwise)
Vector Algebra - Dr. Ray Kwok
Example – moment arm
4N
40o
x
1m
o
1.5 m
1.5 sin(40o)
o
40
x
o
4N
1.5 m
Στo = (4)[1.5 sin(40o)]
= 3.86 N-m (counter-clockwise)
Vector Algebra - Dr. Ray Kwok
Exercise - 1
Find:
(a)
(b)
(c)
(d)
(e)
(f)
r r
A+ B
r
r
B − 2A
r r
A⋅ B
r r
A× B
r r
A× A
r r
B⋅B
r
A = xˆ − 4 zˆ
r
B = 2 xˆ + yˆ + zˆ
(g) Angle between A and B
(h) Find a vector that is perpendicular to A and B ?
Vector Algebra - Dr. Ray Kwok
Triple vector product
scalar
vector
Ax
r r r
A ⋅ B × C = Bx
Cx
(
)
Ay
Az
By
Cy
Bz
Cz
r r r
r r r
r r r r r r
( A × B) × C = ( A ⋅ C ) B − ( B ⋅ C ) A ≠ A × ( B × C )
(homework)
Vector Algebra - Dr. Ray Kwok
OCC
Orthogonal
Orthogonal Curvilinear Coordinates
êi ⋅ ê j = δij
Curvilinear – coordinate surfaces can be curved
Cartesian, Cylindrical & Spherical coordinates
•Transformation between coordinates
•Line, Area & Volume integral in each coordinate
Vector Algebra - Dr. Ray Kwok
Rectangular Polar (2D)
Polar to rectangular
x = r cos θ
y = r sin θ
Rectangular to polar
r2 = x2 + y2
tan θ = y/x
(Pythagorean theorem)
(be certain which angle is θ)
Vector Algebra - Dr. Ray Kwok
Transformation
Vector Algebra - Dr. Ray Kwok
Vector operations
Vector Algebra - Dr. Ray Kwok
Homework
Ch.2 - 3, 5, 10, 12, 13, 15, 20, 23, 26, 30,
and prove eqn-2.33
Also prove that
r r r
r r r
r r r
( A × B) × C = ( A ⋅ C ) B − ( B ⋅ C ) A
r r
r r
r r r r
r r r r
( A × B ) ⋅ (C × D) = ( A ⋅ C )( B ⋅ D ) − ( A ⋅ D )( B ⋅ C )