Complex number

Standard form

Polar form

Exponential form22 yxr

z ( x, y )

r

z = re i θ

z = r (cosθ + i sinθ)

z = x + y i θ

x

y

x

y

Argand Diagram

O

],(arctan

x

yArgz

Operations on TI-89

Conjugate: conj( Real and imaginary parts: real(, imag( Modulus: abs( Principal argument: angle( Solve an equation: cSolve(

azn+bzn-1+…+cz+d=e

Vector Elementary

Two quantities in one

Displacement along an axis (vector in one dimension)

PQ PQ xxPQ

-2 -1 0 1 2 3 4 5

P Q

x

Magnitude:

Direction: from P to Q (positive x)

QP Magnitude:

Direction: from Q to P (negative x)

QP xxQP

QPPQ

QPPQ

Displacement of Q from P = xQ-xP=3-(-2)=5

Displacement of P from Q = xP-xQ=-2-3=-5

Vector on a plane (two-dimensional)

PQa

P (xp, yP)

Q (xQ, yQ)

x

y

yQ

yP

xP xQ

a

= ( ax , ay )

= ( xQ – xP, yQ – yP )

ax

ay

22 )()( PQPQ yyxx

PQa

Magnitude:

(component form)

Right-handed axes

x

z

y90o

O

y

x

z

z

x

y

Left-handed

P(2, 3, 3)

21

O

1 2 31

2

3

Vector in space (three-dimensional)PQa

QPb

ba

z

x

y2

1O

1 2 31

2

3

8

7

6

P(2, 3, 3)

Q(7, 8, 6)a

= (ax, ay, az) = (xQ-xP , yQ-yP , zQ-zP)

b

= (bx, by, bz) = (xP-xQ , yP-yQ , zP-zQ)

222 )()()( PQPQPQ zzyyxx

ba

Same magnitude:

Opposite directions

Vector Operations Equality of two vectors

a = b Same magnitude and direction

OR a = b ax=bx , ay=by , az=bz

z

x

y2

1O

1 2 31

2

3

34

a

b 2

3

Vector Operations Multiplication by a number

ka = ( kax , kay , kaz )

SO -a = ( -ax , -ay , -az )z

x

y2

1O

1 2 31

2

3

34

a

- a 2

3

Vector Operations Commutative Law for Multiplication

Associative Law for Multiplication

Distributive Law for Multiplication

ka = ( kax , kay , kaz )=ak

(m+n)a = ( (m+n)ax , (m+n)ay , (m+n)az )

= ma+na

k(a + b) = ( k(ax+ bx) , k(ay+ by) , k(az + bz) )

=ka + kb

Vector Operations Addition and subtraction

c = a + b = ( ax + bx , ay + by , az + bz )

z

x

yO

a

b

c

z

x

yO

a

b

c

The triangle rule The parallelogram rule

Vector Operations

Commutative Law and Associative Law for addition

z

x

yO

a b

a + b

z

x

yO

ab c

= b + a =(a + b) + c

b + a

(b +c)+a

b + c

a + b(a +b)+c

a + b a + ( b + c )

Application—Relative velocity

vBW

vWS

vBW

vWS

vBS

vBW – velocity of Boat relative to the water

vBS = vBW + vWS

vBW

River

Application—Relative velocityvMP – velocity of man relative to plane

vPg

vmp vPG

vMP

vMG

vMG = vMP+ vPG

Example Problems

A boat is moving across a river. The velocity of flow in the river is 1m/s. And the velocity of the boat with respect to the flow is 2m/s. The river is 500m wide. What is the position of the boat after crossing the river?

Example Problems

The captain of a boat at night can tell that it is moving relative to the sea with a velocity of (3, 2)km/hr, and by observation of lights on the shore its true velocity is found to be

(6, -1)km/hr. What is the velocity of the current?

Example Problems

The pilot of an airplane notes tht the velocity of the plane with respect to the air is 300km/hr due north. From the control tower on the ground, the plane is observed to be flying at 310km/hr with a heading of 7o west of north. Determine the speed and heading of the airflow (wind) with respect to the ground.

Example Problems

x

y

z