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Complex number

Complex number. y. z ( x , y ). y. Standard form Polar form Exponential form. r. z = x + y i. θ. O. x. x. z = r (cos θ + i sin θ ). z = r e i θ. Argand Diagram. Operations on TI-89. Conjugate: conj( Real and imaginary parts: real(, imag( Modulus: abs(

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Complex number

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  1. Complex number y z ( x, y ) y • Standard form • Polar form • Exponential form r z = x + y i θ O x x z = r (cosθ+ isinθ) z = re i θ Argand Diagram

  2. 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

  3. Vector Elementary Two quantities in one

  4. Displacement along an axis (vector in one dimension) P Q -2 -1 0 1 2 3 4 5 x Magnitude: Direction: from P to Q (positive x) Displacement of Q from P = xQ-xP=3-(-2)=5 Magnitude: Direction: from Q to P (negative x) Displacement of P from Q = xP-xQ=-2-3=-5

  5. Vector on a plane (two-dimensional) y (component form) Q (xQ, yQ) = ( ax , ay ) yQ ay a = ( xQ – xP, yQ – yP ) yP Magnitude: P (xp, yP) ax xP xQ x

  6. Right-handed axes Left-handed y z z x O z 90o y 3 x P(2, 3, 3) 2 1 1 2 3 y 1 O 2 x

  7. Vector in space (three-dimensional) = (ax, ay, az) = (xQ-xP , yQ-yP , zQ-zP) z 6 = (bx, by, bz) = (xP-xQ , yP-yQ , zP-zQ) Q(7, 8, 6) a 3 P(2, 3, 3) b 2 1 1 2 3 8 y 1 O 2 Same magnitude: 7 x Opposite directions

  8. Vector Operations • Equality of two vectors a = b Same magnitude and direction ax=bx , ay=by , az=bz a = b OR z 3 2 a 1 3 1 2 3 y 1 2 O b 2 3 4 x

  9. Vector Operations • Multiplication by a number ka = ( kax , kay , kaz ) -a = ( -ax , -ay , -az ) SO z 3 2 a 1 3 1 2 3 y 1 2 O - a 2 3 4 x

  10. 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

  11. Vector Operations • Addition and subtraction c = a + b = ( ax + bx , ay + by , az + bz ) z z b b a a c c y y O O The triangle rule The parallelogram rule x x

  12. Vector Operations • Commutative Law and Associative Law for addition (b +c)+a z z b + c a b + a b c a + b b (a +b)+c a a + b y y O O a + b = b + a a + ( b + c ) =(a + b) + c x x

  13. vBW vBW Application—Relative velocity vBW – velocity of Boat relative to the water vBS = vBW + vWS River vBW vWS vBS vWS

  14. Application—Relative velocity vMP – velocity of man relative to plane vMG = vMP+ vPG vPg vmp vPG vMG vMP

  15. 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?

  16. 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?

  17. 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.

  18. Example Problems

  19. z y x

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