1.
INNER PRODUCT
SPACE
REPORTER: MARK A. SOLIVA

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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
𝑳𝑬𝑵𝑮𝑻 𝑰𝑵 𝑹𝒏

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.
THE EUCLIDEAN
PLANE 𝐸2
MATH 208 – MODERN GEOM

15.
At the end of the lesson the students will be able to:
 Define Euclidean plane.
 Solve the distance between two vectors.
OBJECTIVES

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.
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.
d(P, Q)=|Q-P|

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.
d(P, Q)=|Q-P|
LET
1. P(3, 7) Q(-1, 4)
2. P(1, 2) Q(1, 2)

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.
LETd(P, Q)=|Q-P| d(Q, P)=|P-Q|
1. P(5,-3) Q(-2,4)

23.
𝑳𝑬𝑵𝑮𝑻𝑯 𝑰𝑵 𝑹𝒏
Seatwork: Answer the Following:
1. v= (3, -1, 0)
2. v=(0,1,2,3)

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.
ASSIGNMENT

26.
Thank You!