Chapter 5 �Dot, Inner and Cross Products
5.1 Length of a vector
5.2 Dot Product
5.3 Inner Product
5.4 Cross Product
5.1 Length and Dot Product in Rn
The length of a vector in Rn is given by
is called a unit vector.
2/45
(a) In R5, the length of is given by
(b) In R3 the length of is given by
(v is a unit vector)
3/45
u and v have the same direction
u and v have the opposite direction
the standard unit vector in R2:
the standard unit vector in R3:
4/45
Let v be a vector in Rn and c be a scalar. Then
Pf:
5/45
If v is a nonzero vector in Rn, then the vector
has length 1 and has the same direction as v. This vector u is called the unit vector in the direction of v.
Pf:
v is nonzero
(u has the same direction as v)
(u has length 1 )
6/45
(1) The vector is called the unit vector in the direction of v.
(2) The process of finding the unit vector in the direction of v
is called normalizing the vector v.
7/45
Find the unit vector in the direction of ,
and verify that this vector has length 1.
is a unit vector.
Sol:
8/45
The distance between two vectors u and v in Rn is
(1)
(2) if and only if
(3)
9/45
The distance between u=(0, 2, 2) and v=(2, 0, 1) is
10/45
5.2 Dot Product
The dot product of and
is the scalar quantity
The dot product of u=(1, 2, 0, -3) and v=(3, -2, 4, 2) is
11/45
If u, v, and w are vectors in Rn and c is a scalar,
then the following properties are true.
(1)
(2)
(3)
(4)
(5) , and if and only if
12/45
Rn was defined to be the set of all order n-tuples of real numbers. When Rn is combined with the standard operations of vector addition, scalar multiplication, vector length, and the dot product, the resulting vector space is called Euclidean n-space.
13/45
Sol:
14/45
Given
Sol:
Find
15/45
Verify the Cauchy - Schwarz inequality for u=(1, -1, 3)
and v=(2, 0, -1)
If u and v are vectors in Rn, then
( denotes the absolute value of )
Sol:
16/45
The angle between the zero vector and another vector is not defined.
Opposite
direction
Same
direction
17/45
Sol:
u and v have opposite directions.
18/45
Two vectors u and v in Rn are orthogonal if
The vector 0 is said to be orthogonal to every vector.
19/45
Determine all vectors in Rn that are orthogonal to u=(4, 2).
Let
Sol:
20/45
If u and v are vectors in Rn, then
Pf:
Equality occurs in the triangle inequality if and only if
the vectors u and v have the same direction.
21/45
If u and v are vectors in Rn, then u and v are orthogonal
if and only if
22/45
(A vector in Rn
is represented as an n×1 column matrix)
23/45
5.3 Inner Product
Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number <u, v> with each pair of vectors u and v and satisfies the following axioms.
(1)
(2)
(3)
(4) and if and only if
24/45
A vector space V with an inner product is called an inner product space.
Vector space:
Inner product space:
25/45
Show that the dot product in Rn satisfies the four axioms of an inner product.
Sol:
By Theorem 5.3, this dot product satisfies the required four axioms. Thus it is an inner product on Rn.
26/45
Show that the function defines an inner product on R2, where and .
Sol:
27/45
28/45
Show that the following function is not an inner product on R3.
Sol:
Let
Axiom 4 is not satisfied.
Thus this function is not an inner product on R3.
29/45
Let u, v, and w be vectors in an inner product space V, and let c be any real number.
(1)
(2)
(3)
30/45
u and v are orthogonal if .
31/45
(1) If , then v is called a unit vector.
(2)
(the unit vector in the
direction of v)
not a unit vector
32/45
is an inner product
Sol:
33/45
(1)
(2) if and only if
(3)
(1)
(2) if and only if
(3)
34/45
Let u and v be vectors in an inner product space V.
(1) Cauchy-Schwarz inequality:
(2) Triangle inequality:
(3) Pythagorean theorem :
u and v are orthogonal if and only if
Theorem 5.5
Theorem 5.6
Theorem 5.4
35/45
Let u and v be two vectors in an inner product space V, such that . Then the orthogonal projection of u onto v is given by
If v is a init vector, then .
The formula for the orthogonal projection of u onto v takes the following simpler form.
36/45
Use the Euclidean inner product in R3 to find the orthogonal projection of u=(6, 2, 4) onto v=(1, 2, 0).
Sol:
37/45
Let u and v be two vectors in an inner product space V, such that . Then
38/45
5.4 Cross Product
The cross product of and
is the vector quantity
The cross product of u=(1, 2, 0) and v=(3, -2, 4) is
39/45
Let u, v and w be 3 vectors in R3, then:
Let u, v and w be 3 vectors in R3 and k a scalar, then:
40/45
Let u, v and w be 3 vectors in R3, then:
41/45