1 / 19

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(

vinny
Télécharger la présentation

Complex number

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related