The Norm and Dot Product

Norm §

The norm (aka length or magnitude) of = || =

= then | | = =

Properties of the Norm §

Let and c . Then

1. \| | 0 with equality if and only if
2. \| c | = | c | | |
3. \| | | | + | | (The Triangle Inequality)

2 nonzero vectors in are parallel if they are scalar multiples of one other

A vector is a unit vector if | | = 1

is called the normalization of

Dot Product §

Let = and = be vectors in . The dot product of and is the real number

=

Properties of Dot Product §

Let and c

2. =
3. = 0
4. = | |

Cauchy–Schwarz Inequalty §

This is how the theorem is spelled in the textbook

For any two vectors we have

|

Two vectors are said to be orthogonal if

= arccos (

• and determine an acute angle
• and are perpendicular
• and determine an obtuse angle