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Understanding Coordinate Systems and Vector Operations in 2D and 3D Geometry

This guide covers the fundamentals of basic mathematics for geometric modeling, focusing on coordinate reference frames including Cartesian and polar coordinates in 2D, and extending to 3D Cartesian coordinates. It will explore the definitions and properties of geometric objects like points and vectors, including vector magnitude, direction, addition, scalar multiplication, and the dot and cross products. By learning how to convert between polar and Cartesian coordinates and manipulate vectors, readers will gain essential skills for geometric analysis and modeling.

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Understanding Coordinate Systems and Vector Operations in 2D and 3D Geometry

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  1. Basic mathematics for geometric modeling

  2. Coordinate Reference Frames y • Cartesian Coordinate (2D) • Polar coordinate (x, y) x r 

  3. Y P P y r r y   x x x Relationship : polar & cartesian Use trigonometric, polar  cartesian x = r cos  , y = r sin  Cartesian  polar r =  x2 + y2,  = tan-1 (y/x)

  4. 3D cartesian coordinates y y x x z z Right-handed 3D coordinate system

  5. POINT • The simplest of geometric object. • No length, width or thickness. • Location in space • Defined by a set of numbers (coordinates) e.g P = (x, y) or P = (x, y, z) • Vertex of 2D/ 3D figure

  6. VECTOR • distance and direction • Does not have a fixed location in space • Sometime called “displacement”.

  7. VECTOR • Can define a vector as the difference between two point positions. y y2 V Q y1 P x x2 x1 V = Q – P = (x2 – x1, y2 – y1) = (Vx, Vy) Also can be expressed as V = Vxi + Vyj Component form

  8. VECTOR : magnitude & direction • Calculate magnitude using the Pythagoras theorem  distance • |V| = Vx2 + Vy2 • Direction •  = tan-1 (Vy/Vx)

  9. VECTOR : magnitude & direction V Q • Example 1 • If P(3, 6) and Q(6, 10). Write vector V in component form. • Answer • V = [6 - 3, 10 – 6] = [3, 4]

  10. VECTOR : magnitude & direction • Example 1 (cont) • Compute the magnitude and direction of vector V • Answer • Magnitud |V| = 32 + 42 • = 25 = 5 • Direction  = tan-1 (4/3) = 53.13

  11. Unit Vector • As any vector whose magnitude is equal to one • V = V |V| • The unit vector of V in example 1 is = [Vx/|V| , Vy/|V|] = [3/5, 4/5]

  12. VECTOR : 3D y • Vector Component • (Vx, Vy, Vz) • Magnitude • |V| = Vx2 + Vy2 + Vz2 • Direction •  = cos-1(Vx/|V|),  = cos-1(Vy/|V|),  =cos-1(Vz/|V|) • Unit vector • V = V = [Vx/|V|, Vy/|V|, Vz/|V|] |V| Vy V x Vz Vx z

  13. Scalar Multiplication • kV = [kVx, kVy, kVz] • If k = +ve  V and kV are in the same direction • If k = -ve  V and kV are in the opposite direction • Magnitude |kV| = k|V|

  14. Scalar Multiplication • Base on Example 1 • If k = 2, find kV and the magnitudes • Answer • kV = 2[3, 4] = [6, 8] • Magnitude |kV|= 62 + 82 = 100 = 10 • = k|V| = 2(5) = 10

  15. Vector Addition y y • Sum of two vectors is obtained by adding corresponding components • U = [Ux, Uy, Uz], V = [Vx, Vy, Vz] • U + V = [Ux + Vx, Uy + Vy, Uz + Vz] V V U + V U U x x

  16. P Q Vector Addition Q • Example • If vector P=[1, 5, 0], vector Q=[4, 2, 0]. Compute P + Q • answer • P + Q = [1+4, 5+2, 0+0] = [5, 7, 0] P

  17. Vector Addition & scalar multiplication properties • U + V = V + U • T + (U + V) = (T + U) + V • k(lV) = klV • (k + l)V = kV + lV • k(U + V) = kU + kV

  18. Scalar Product • Also referred as dot product or inner product • Produce a number. • Multiply corresponding components of the two vectors and add the result. • If vector U = [Ux, Uy, Uz], vector V = [Vx, Vy, Vz] • U . V = UxVx + UyVy + UzVz

  19. Scalar Product. • Example • If vector P=[1, 5, 0], vector Q=[4, 2, 0]. Compute P . Q • answer • P . Q = 1(4) + 5(2) + 0(0) • = 14

  20. U V Scalar Product properties • U.V = |U||V|cos  • angle between two vectors •  =cos –1(U.V) • |U||V| • Example • Find the angle between vector b=(3, 2) and vector c = (-2, 3) 

  21. Scalar Product properties Solution • b.c = (3, 2). (-2, 3) • 3(-2) + 2(3) = 0 • |b| = 32 + 22 = 13 = 3.61 • |c| = (-2)2 + 32 = 13 = 3.61 •  =cos –1 ( 0/(3.61((3.61)) •  = cos –1 ( 0 ) = 90

  22. Scalar Product properties • If U is perpendicular to V, U.V = 0 • U.U = |U|2 • U.V = V.U • U.(V+W) = U.V + U.W • (kU).V = U.(kV)

  23. Vector Product • Also called the cross product • Defined only for 3 D vectors • Produce a vector which is perpendicular to both of the given vectors. y c = a x b c x b a z

  24. Vector Product • To find the direction of vector C, use righ-hand rules C x B x B A z C A z

  25. B x A A x B C x x z B A z C B A Vector Product • To find the direction of vector C, use righ-hand rules

  26. exercise • Find the direction of vector C, (keluar skrin atau kedalam skrin) A x B Q B P x Q P A O M x N L x O M L N

  27. Vector Product • If vektor A = [Ax, Ay, Az], vektor B = [Bx, By, Bz] • A x B = i j k i j • Ax Ay Az Ax Ay • Bx By Bz Bx By = [ (AyBz-AzBy), (AzBx-AxBz), (AxBy-AyBx)]

  28. Vector Product P • Example • If P=[1, 5, 0], Q=[4, 2, 0]. Compute P x Q • Solution • P x Q = i j k i j • 1 5 0 1 5 • 4 2 0 4 2 • = [ (5.(0)-0.(5)), (0.(4)-1.(0)), (1.(2)-5.(4))] • = [ 0, 0, -18] Q

  29. Vector Product • Properties • U x V = |U||V|n sin  where n = unit vector perpendicular to both U and V • U x V = -V x U • U x (V + W) = U x V+ U x W • If U is parallel to V, U x V = 0 • U x U = 0 • kU x V = U x kV

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