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# INNER_SPACE_PRODUCT-EUCLIDEAN_PLANE.pptx

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# INNER_SPACE_PRODUCT-EUCLIDEAN_PLANE.pptx

modern geometry

modern geometry

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### INNER_SPACE_PRODUCT-EUCLIDEAN_PLANE.pptx

1. 1. INNER PRODUCT SPACE REPORTER: MARK A. SOLIVA
2. 2. At the end of the lesson the students will be able to:  Determine whether a function defines an inner product.  Find the inner product of two vector in 𝑅𝑛. OBJECTIVES
3. 3. INNER-PRODUCT SPACE It is represented by angle brackets <u, v> It is a vector space with additional structure called an inner product. It is a function that associate a real number <u, v> that satisfies the following axioms…
4. 4. PROPERTIES OF INNER PRODUCT  Commutative Property of the Inner Product ( <u, v> = (v, u> )  Distributive Property of the Inner Product (<u, v+w>=<u, v>+<u, w>)  Associative Property of Inner Product ( k<u, v>=<ku, v>)  Positive Semi-Definite < 𝑣, 𝑣 > ≥ 0 if and only if v = 0  Point-Separate/Non-Degenerate < 𝑣, 𝑣 > = 0 if and only if v = 0
5. 5. Commutative Property of the Inner Product (<u, v> = <v, u>) Example 1: 𝑢 = 𝑢1, 𝑢2, … 𝑢𝑛 𝑣 = (𝑣1, 𝑣2, … 𝑣𝑛) Solution: < 𝑢, 𝑣 > = 𝑢1𝑣1 + 𝑢2𝑣2 + ⋯ + 𝑢𝑛𝑣𝑛
6. 6. 1. Commutative Property of the Inner Product (<u, v>=<v , u>) Example 2: u = (2, 3) v = (-1, 2) Solution: (<u, v>=<v, u>) 𝒖𝟏𝒗𝟏 + 𝒖𝟐𝒗𝟐 = 𝒗𝟏𝒖𝟏 + 𝒗𝟐𝒖𝟐 (2*-1)+(3*2)=(-1*2)+(2*3) -2+6=-2+6 4=4
7. 7. Commutative Property of the Inner Product (<u, v> =<v, u>) Example 3: u = (0, -5, 2) v= (6, 9, 0) Solution: < 𝒖, 𝒗 > = 𝒖𝟏𝒗𝟏 + 𝒖𝟐𝒗𝟐 + 𝒖𝟑𝒗𝟑 < 𝒖, 𝒗 > = 𝟎 ∗ 𝟔 + −𝟓 ∗ 𝟗 + 𝟐 ∗ 𝟎 = −𝟒𝟓
8. 8. DISTRIBUTIVE PROPERTY OF THE INNER PRODUCT (<u, v+w>=<u, v>+<u, w>) Example: u= (2, 3) k= 2 v= (-1, 2) w=(3, 1) v+w = (2, 3) Solution: (<u, v+w>=<u, v>+<u, w>) 𝒖𝟏(𝒗 + 𝒘)𝟏+𝒖𝟐(𝒗 + 𝒘)𝟐= 𝒖𝟏𝒗𝟏 + 𝒖𝟐𝒗𝟐 + 𝒖𝟏𝒘𝟏 + 𝒖𝟐𝒘𝟐 𝟐 ∗ 𝟐 + 𝟑 ∗ 𝟑 = 𝟐 ∗ −𝟏 + 𝟑 ∗ 𝟐 + 𝟐 ∗ 𝟑 + 𝟑 ∗ 𝟏 𝟒 + 𝟗 = −𝟐 + 𝟔 + 𝟔 + 𝟑 𝟏𝟑 = 𝟏𝟑
9. 9. DISTRIBUTIVE PROPERTY OF THE INNER PRODUCT (<u, v+w>=<u, v>+<u, w>) EXAMPLE 2: u=(0, -1) w=(-1,2) v=(2,-3) v+w=(1,-1) Solution: (<u, v+w>=<u, v>+<u, w>) 𝒖𝟏(𝒗 + 𝒘)𝟏+𝒖𝟐(𝒗 + 𝒘)𝟐= 𝒖𝟏𝒗𝟏 + 𝒖𝟐𝒗𝟐 + 𝒖𝟏𝒘𝟏 + 𝒖𝟐𝒘𝟐 𝟎 ∗ 𝟏 + −𝟏 ∗ −𝟏 = 𝟎 ∗ 𝟐 + −𝟏 ∗ −𝟑 + 𝟎 ∗ −𝟏 + −𝟏 ∗ 𝟐 𝟎 + 𝟏 = 𝟎 + 𝟑 + 𝟎 + (−𝟐) 𝟏 = 𝟏
10. 10. ASSOCIATIVE PROPERTY OF INNER PRODUCT (k<u, v>=<ku, v>) Example : 1 u=(2, 3) k=2 v=(-1, 2) ku=(4,6) Solution: 𝒌 < 𝒖, 𝒗 > = < 𝒌𝒖, 𝒗 > 𝒌 𝒖𝟏𝒗𝟏 + 𝒖𝟐𝒗𝟐 = 𝒌𝒖𝟏𝒗𝟏 + 𝒌𝒖𝟐𝒗𝟐 𝟐 𝟐 ∗ −𝟏 + 𝟑 ∗ 𝟐 = 𝟒 ∗ −𝟏 + 𝟔 ∗ 𝟐 𝟐 −𝟐 + 𝟔 = −𝟒 + 𝟏𝟐 𝟖 = 𝟖
11. 11. ASSOCIATIVE PROPERTY OF INNER PRODUCT (k<u, v>=<ku, v>) Example 2: u=(0, 1) k=3 v=(2,3) ku=(0,3) Solution: 𝒌 < 𝒖, 𝒗 > = < 𝒌𝒖, 𝒗 > 𝒌 𝒖𝟏𝒗𝟏 + 𝒖𝟐𝒗𝟐 = 𝒌𝒖𝟏𝒗𝟏 + 𝒌𝒖𝟐𝒗𝟐 𝟑 𝟎 ∗ 𝟐 + 𝟏 ∗ 𝟑 = 𝟎 ∗ 𝟐 + 𝟑 ∗ 𝟑 𝟑 𝟎 + 𝟑 = 𝟎 + 𝟗 𝟗 = 𝟗
12. 12. 𝑳𝑬𝑵𝑮𝑻 𝑰𝑵 𝑹𝒏
13. 13. 𝑳𝑬𝑵𝑮𝑻𝑯 𝑰𝑵 𝑹𝒏 Example 1: v=(0, -2, 1, 4, -2) 𝒗 = 𝟎𝟐 + −𝟐𝟐 + 𝟏𝟐 + 𝟒𝟐 + (−𝟐𝟐) = 𝟐𝟓 = 𝟓 Example 2: 𝒗 = 𝟐 𝟏𝟕 , − 𝟐 𝟏𝟕 , 𝟑 𝟏𝟕 𝒗 = 𝟐 𝟏𝟕 𝟐 + − 𝟐 𝟏𝟕 𝟐 + 𝟑 𝟏𝟕 𝟐 = 𝟏𝟕 𝟏𝟕 = 𝟏 (𝑼𝒏𝒊𝒕 𝑽𝒆𝒄𝒕𝒐𝒓)
14. 14. THE EUCLIDEAN PLANE 𝐸2 MATH 208 – MODERN GEOM
15. 15. At the end of the lesson the students will be able to:  Define Euclidean plane.  Solve the distance between two vectors. OBJECTIVES
16. 16. THE EUCLIDEAN PLANE 𝑬𝟐 The plane has both algebraic and geometric aspects. The algebraic properties focuses on the vector properties of 𝑅2. In the geometric properties we will focus on the concept of Distance.
17. 17. THE EUCLIDEAN PLANE 𝑬𝟐 If P and Q are points, we define the distance between P and Q by the equation: d(P, Q) = |𝑸 − 𝑷| The symbol 𝑬𝟐 will be used to denote the set of points in Euclidean plane equipped with the distance function d.
18. 18. d(P, Q)=|Q-P|
19. 19. THE EUCLIDEAN PLANE 𝑬𝟐 𝒅 𝑷, 𝑸 = |𝑸 − 𝑷| Most important properties of the distance: Theorem 5. let P, Q and R be points of 𝑬𝟐, 𝒕𝒉𝒆𝒏 i. d(P,Q)≥ 𝟎 ii. d(P,Q)=0 if and only if P=Q iii. d(P,Q)=d(Q,P) iv. d(P,Q)+d(Q,R)≥ 𝒅(𝑷, 𝑹)
20. 20. d(P, Q)=|Q-P| LET 1. P(3, 7) Q(-1, 4) 2. P(1, 2) Q(1, 2)
21. 21. d(P, Q)=|Q-P| LET d(Q, R)=|R-Q| d(P, R)=|R-P| 1. P(1, 5) Q(-2.1) R(3,-2)
22. 22. LETd(P, Q)=|Q-P| d(Q, P)=|P-Q| 1. P(5,-3) Q(-2,4)
23. 23. 𝑳𝑬𝑵𝑮𝑻𝑯 𝑰𝑵 𝑹𝒏 Seatwork: Answer the Following: 1. v= (3, -1, 0) 2. v=(0,1,2,3)
24. 24. ASSESSMENT Answer the following: A. Given: u= (2, -2) v=(5,8) w=(-4, 3) 1. <u, v> 2. ||𝒗|| 3. <u, v+w> B. Find the of the vector
25. 25. ASSIGNMENT
26. 26. Thank You!