Chapter 2

Vector Algebra

Review

Dr. Ray Kwok

SJSU

Vector Algebra - Dr. Ray Kwok

Vector products

BArr⋅

(scalar) (scalar) = scalar, (a)(b) = ab

e.g. 2(4 kg) = 8 kg

(scalar) (vector) = vector,

e.g.

(scalar) (scalar) = scalar

e.g. 2(4 kg) = 8 kg

(vector) times (vector) = ?

can be scalar (scalar product, or dot product)

or

vector (vector product, or cross product) BArr

×

Can you add a vector to a scalar?

( ) AkAkrr

=

( ) yxyx ˆ20ˆ10ˆ4ˆ25 +=+

Vector Algebra - Dr. Ray Kwok

( ) ( )

zzyyxx

zyxzyx

BABABABA

zByBxBzAyAxABA

zyzx0yx

zzyy1xx

cosABBA

++=⋅

++⋅++=⋅

⋅=⋅==⋅

⋅=⋅==⋅

θ=⋅

rr

rr

rr

e.g.

yx3B

y2x2A

+=

−=r

r

What is A · B ?

e.g.

What’s the

angle between

these 2 arrows?

The scalar product (dot product)

Vector Algebra - Dr. Ray Kwok

Interpretation - projection

Vector Algebra - Dr. Ray Kwok

Example - Work

xFWrr

∆⋅=

So, is the uplifting force doing anything at all??

(Projection)

Vector Algebra - Dr. Ray Kwok

yzx

xyz

zxy

yxz

xzy

zyx

zzyy0xx

sinABBA

−=×

−=×

−=×

=×

=×

=×

×=×==×

θ=×rr

e.g.

yx3B

y2x2A

+=

−=r

r

What is A x B ?

e.g.??

What’s the

angle between

these 2 arrows?

zyx

zyx

BBB

AAA

zyx

BA ≡×rr

right-hand coordinate

cyclic permutation

The vector product (cross product)

Vector Algebra - Dr. Ray Kwok

Example – calculate torque

o

x

4 N

40o

1.5 m

Στo = (1.5)(4)sin(40o)

= 3.86 N-m (counter-clockwise)

choose “+” = counter-clockwise

Frrrr

×=τ

Vector Algebra - Dr. Ray Kwok

Example – perpendicular F

o

x

4 N

40o

1.5 m

Στo = (1.5)[4 sin(40o)]

= 3.86 N-m (counter-clockwise)

o

x

4 sin(40o)

1.5 m

Vector Algebra - Dr. Ray Kwok

Example – moment arm

o

x

4 N

40o

1 m 1.5 m

Στo = (4)[1.5 sin(40o)]

= 3.86 N-m (counter-clockwise)

o

x

4 N40

o

1.5 m

1.5 sin(40o)

Vector Algebra - Dr. Ray Kwok

Exercise - 1

Find:

(a)

(b)

(c)

(d)

(e)

(f)

(g) Angle between A and B

(h) Find a vector that is perpendicular to A and B ?

BB

AA

BA

BA

AB

BA

rr

rr

rr

rr

rr

rr

⋅

×

×

⋅

−

+

2

zyxB

zxA

ˆˆˆ2

ˆ4ˆ

++=

−=r

r

Vector Algebra - Dr. Ray Kwok

Triple vector product

( )zyx

zyx

zyx

CCC

BBB

AAA

CBA =×⋅rrrscalar

(homework)

)()()()( CBAACBBCACBArrrrrrrrrrrr

××≠⋅−⋅=××vector

Vector Algebra - Dr. Ray Kwok

OCC

ijji ee δ=⋅Orthogonal

Curvilinear – coordinate surfaces can be curved

•Transformation between coordinates

•Line, Area & Volume integral in each coordinate

Cartesian, Cylindrical & Spherical coordinates

Orthogonal Curvilinear Coordinates

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 (Pythagorean theorem)

� tan θ = y/x (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

ACBBCACBArrrrrrrrr)()()( ⋅−⋅=××

Ch.2 - 3, 5, 10, 12, 13, 15, 20, 23, 26, 30,

and prove eqn-2.33

Also prove that ))(())(()()( CBDADBCADCBArrrrrrrrrrrr

⋅⋅−⋅⋅=×⋅×