MAT208 FALL 2009

1. MAT208 FALL 2009 Sections 5.1-5.3 Kolman/Hill

2. Length and Direction in R2 and R3 • Let be a vector in R2. The length or magnitude of the vector, denoted by is Also called norm. • Let and be vectors in R2. The distance between u and v is defined as the magnitude of u - v, i.e. as

3. Length and Direction in R2 and R3 • Concepts can be extended to R3

4. Angles • Find the angle between two vectors Lengths of sides are , and

5. Angles By the law of cosines An angle can be computed from the coordinates of the vectors

6. Angles • For vectors and in R2

7. Direction Cosines • In R3, let and let be the natural basis for R3. Let a be the angle between v and the x-axis, b be the angle between v and the y-axis and c be the angle between v and the z-axis. The quantities cos a, cos b and cos are called the direction cosines of v and can be computed from v and i, j, k .

8. Inner Product • Let and be vectors in R2. The inner product or dot product of u and v, denoted by ( u,v ) or u•v, is u•v=u1v1 + u2v2. Similarly, for and in R3, u•v=u1v1 + u2v2 + u3v3

9. Orthogonality • Two vectors u and v in R2 or R3 are orthogonal or perpendicular if u•v= 0. • Theorem - Let u, v and w be vectors in R2 or R3. Then 1) u•u > 0 if u ≠ 0, and u•u= 0 if and only if u=0 2) u•v=v•u 3) (u + v)•w=u•w + v•w 4) (cu)•v=c(u•v), for any real scalar c

10. Unit Vectors • A unit vector in R2 or R3 is a vector whose length is 1. • Note - If x is a nonzero vector, then the vector is a unit vector .

11. Cross Product • Let u, vR3 . Try to find a vector w that is orthogonal to both u and v, i.e. u • w= 0, v • w= 0

12. Cross Product • Two equations in the three unknowns x, y and z. No unique solution for x, y and z. • One solution is x=u2v3 - u3v2, y=u3v1 - u1v3, z=u1v2 - u2v1 .

13. Cross Product in R3Orthogonality • Note that Similar calculation shows that v • w= 0

14. Properties of the Cross Product • u x v= - v x u • ux (v + w) =u x v + u x w • (u + v) x w=u x w + v x w • c (u x v) = (cu) x v=u x (cv) • u x u= 0 • 0 x u= u x 0= 0 • u x (v x w) = (u • w) v - (u • v) w • (u x v) x w= (w • u) v - (w • v) u

15. Cross Products of Natural Basis Vectors Cross Products of Natural Basis Vectors Direction of w=u x v determined by right hand rule.

16. Angles Again • An angle between two vectors can also be determined by using the cross product. We will need the following identity: (u x v) • w = u • (v x w)

17. Angles (continued)

18. Cross Product in R2 ?? • If u and v are in R2 then define cross product for them by considering This is not a vector in R2, but it is still useful: Consider complex numbers u1 + iu2 and v1 + iv2 and compute (u1 - iu2)(v1 + iv2) = (u1 v1 + u2v2) + i(u1v2 -u2 v1) Real part is dot product and imaginary part is cross product.

19. Determinants and Cross Product • Let u= u1 i + u2 j + u3 k, v= v1 i + v2 j + v3 k. Then u x v= (u2v3 – u3v2)i + (u3v1 – u1v3)j + (u1v2 – u2v1)k

20. Inner Product Spaces • Let V be a real vector space. An inner product on V is a mapping from V x V to R that assigns, to each ordered pair of vectors u and v of V, a real number ( u,v ) satisfying: a) (u,u) > 0 for u ≠ 0, and (u,u) = 0 if and only if b) (u,v) = (v,u) for all u, v in V c) (u+v,w) = (u,w) + (v,w) for all u, v, w in V d) (cu,v) =c(u,v) for all u, v in V and for all real c u=0

21. Inner Product Spaces • Example - The standard inner product on Rn can be defined as (u,v) =u1v1 + u2v2 +… unvn where

22. Inner Product Spaces • Example - Let and be vectors in R2. • Let ( u,v ) =u1v1 - u2v1 - u1v2 + 3u2v2. Show that this gives an inner product on R2 • Let ( u,u ) = 0 then u1=u2 and u2= 0. So u=0 If u=0 , then ( u,u ) = ( 0 - 0 ) 2 + 2 • 0 2= 0

23. Inner Product Spaces Example (continued) b) c) d)

24. Inner Product Spaces • Let V be the vector space C[0,1] of all continuous functions on the interval [0,1]. For f,g V define • From calculus if f ≠ zero function. Also, ( f,f) = 0 if and only if f (t) º 0.

25. Inner Product Spaces b) c) d)

26. Positive Definite Matrix • An nxn symmetric matrix A with the property that xTAx > 0 for every nonzero vector xRn is called positive definite . • A positive definite matrix is nonsingular. To see this, suppose that A is positive definite but A is singular. Then Ax=0 has a nontrivial solution x0. Then x0TAx0=0, which contradicts the positive definite nature of A. So A is nonsingular.

27. Definitions • A real vector space that has an inner product defined on it is called an inner product space. If the space is finite dimensional, it is called a Euclidean space. • In an inner product space, the length of a vector u is defined as • Since (0,0) = 0, we have

28. Cauchy-Schwarz Inequality • Theorem (Cauchy-Schwarz Inequality) - If u and v are any two vectors in an inner product space V, then • Proof - Let u and v be arbitrary vectors in V. If u=0, then and (u,v) = 0 and the inequality holds. So, suppose u ≠ 0. Let r be a scalar and consider the vector ru + v. Then where a = (u,u), b = 2(u,v) and c = (v,v).

29. Cauchy-Schwarz (continued) • Proof (continued) Since ar2 + br + c ≥ 0, the equation ar2 + br + c= 0 has at most one real solution. Thus b2 - 4ac ≤ 0 or b2 ≤ 4ac, which implies that 4(u,v)2 ≤ 4(u,u)(v,v), i.e. QED

30. Inner Product Spaces Example • In R3, let and Using the standard inner product in R3

31. The Triangle Inequality • Theorem (Triangle Inequality) - Let u and v be vectors in an inner product space V. Then Proof By the Cauchy-Schwarz inequality:

32. The Triangle Inequality (continued) Thus QED

33. More Definitions • Let V be an inner product space and let u V and v V. The distance between u and v is • Let V be an inner product space and let u V and v V. Then, u and v are orthogonal if ( u,v ) = 0

34. Inner Product Spaces • Let V = P2, the set of all polynomials of degree ≤ 2. For p(t), q(t)P2 define an inner product then the vectors t and t - 2/3 are orthogonal since

35. Orthonormal • Let V be an inner product space. A set S of vectors in V is called orthogonal if every two distinct vectors in S are orthogonal. If every vector is S also has unit length, then S is orthonormal . • If x is a nonzero vector in an inner product space, one can always find a unit vector u in the same direction as x .