Monday, January 13 − Lecture 4 : Dot products and norms
1. dot product of two vectors in ℝ .
2. algebraic properties of dot product of two vectors.
3. length or norm of a vector in ℝ . n
4. unit vector in the direction of a n
5. angle between two vectors in ℝ .
6. distance between two points in ℝ . n
7. Cauchy-Schwartz inequality
8. orthogonal vectors or ortnogonal sets in ℝ .
9. orthonormal sets in ℝ .
4.1 Dot product operation on ℝ definition.
Operation What we start with : What we get :
n 1 number x⋅y in ℝ :
2 vectors x and y in ℝ : x⋅y = x y + x y + ..., x y .
x = (x 1 x2, ...,nx ) 1 1 2 2 n n
Dot product and Other notation for dot-product:
y = ( y , y , ...,y ). x⋅y = < x, y >
1 2 n
4.1.1 Examples − If x = (2, 1, 7) and y = (3, − 9, 0) verify that x ⋅ y = −3.
4.1.2 Theorem − Algebraic properties of dot product. Suppose x, y and z are in ℝ
and α in ℝ. Then
DP1 x ⋅x ≥ 0
DP2 x ⋅y = y ⋅x
DP3 (αx) ⋅y = α(x ⋅y)
DP4 x ⋅y + z) = x ⋅y + x ⋅z
Proof is straight forward and is left as an exercise.
Norm or length of a vector
4.2 Definition − For a vector x in ℝ , the length or norm of x, denoted by || x || , is the
4.2.1 Properties of the norm in ℝ .
It can be shown that the norm on ℝ satisfies 3 properties:
N1 For every vector x, || x || ≥ 0. Furthermore, ||x|| = 0 ⇔ x = 0. (Positive definite property)
N2 || αx || = |α| || x || for all scalars α, and x in V. (Homogeneous property)
N3 || x + y || ≤ || x || + || y || for all x, y in V(Triangle inequality property)
4.2.2 Observation − If a is any non-zero vector in ℝ , then the vector
defined as b = (1/|| a ||)a has norm equal to 1 since
The vector a/|| a || is called the unit vector in the direction of a.
Note that if b is a vector which points in the same direction as a then
This is true since if b and a point in the same direction then they both have the same
unit vector, i.e., a / || a || = b / || b ||.
4.2.3 Remark – In ℝ the norm, thus defined, leads to the familiar distance formula
between two points x and y. The distance betweenx = (x , x ) and y = (y , y ) is
1 2 1 2
Definition − Let a and b be two vectors in ℝ . The distance between a and b is
|| b − a ||
4.3 Theorem − Cauchy-Schwartz inequality. For any vectors x and y in ℝ
| x⋅y | ≤ || x || || y ||
Let x, y be two vectors in ℝ and t be a variable scalar. For any real number t: (Be sure to verify the equality between the expression in the f