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VECTORS

VECTORS. Learning Objectives :  Vector Definition  Algebraic Properties of Vectors  Dot and Cross Products. AB. y. B. A. D. x. O. C. y. D(-1,6). B(3,4). v. u. C(-4,2). x. O. v=5 , slope of v is 4/3. u=5 , slope of u is 4/3. .

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VECTORS

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  1. VECTORS Learning Objectives:  Vector Definition  Algebraic Properties of Vectors  Dot and Cross Products

  2. AB

  3. y B A D x O C

  4. y D(-1,6) B(3,4) v u C(-4,2) x O v=5 , slope of v is 4/3. u=5 , slope of u is 4/3. .

  5. Components(bileşenleri) of a vector v=[v1,v2] are v1 and v2 . P(x1,y1) , Q(x2,y2) PQ=[x2-x1, y2-y1]

  6. Norm of a vector is

  7. Algebraic Properties of Vectors Let u=[u1,u2] and v=[v1,v2] be vectors and k scalar. Addition: u+v =[u1+ v1, u2 +v2] ku=[ ku1, ku2 ]

  8. y u-v v -v u+v u x O

  9. Algebraic Properties of Vectors • u+v=v+u 2) (u+v)+w = u+(v+w) • 3) u+0 =u 4) u+ (-u) =0 • 5) 0u =0 6) 1u =u • 7) k(u+v) = ku +kv , k scalar

  10. i=[1,0] and j=[0,1] are unit(birim) vectors v=[3,5 ] =3[1,0]+5[0,1] =3i+5j.

  11. Example: v=3i-4j length direction

  12. y v (Inner or Dot Products)  u x u=[u1,u2] , v=[v1,v2] u.v= u1v1+u2v2 u.v=

  13. Algebraic Properties of Dot Product • 1) u.v=v.u • 2) (ku).v==u.(kv)=k(u.v) • u.(v+w)=u.v+u.w • u.u = u 2 • 5) 0.u =0

  14. (Dik) perpendicular or orthogonal vectors Question: Letu[3,-2] and v=[4,6] be two vectors. Are these vectors orthogonal or not?

  15. z D P(x,y,z) C y O A B x

  16. i=[1,0,0] , j=[0,1,0] and k=[0,0,1] OP= x.i+yj+zk =[x,y,z]. i.i=j.j=k.k=1 ve i.j=j.k=k.i=0

  17. Cross Product of Vectors u=[a,b,c] , v=[d,e,f] uv = [bf-ce , cd-af , ae-bd]

  18. 1) vu = - uv 2) u(v+w) = uv + uw 3) aubv = (ab) uv , a and b scalar 4) ii= jj =kk=0 5) ij=k ; jk=i ; ki=j 6) j i= -k ; kj= -i ; ik= -j

  19. uv is perpendicular to u andv uv v u

  20. Web sites http://math.rice.edu/~lanius/Lessons/index.html http://mathforum.com/dr.math/drmath.college.html İ Y İ Ç A L I Ş M A L A R . . .

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