# Connexions

You are here: Home » Content » Introductory Linear Algebra por Bernard Kolman 8/E

## Navigation

### Recently Viewed

This feature requires Javascript to be enabled.

# Introductory Linear Algebra por Bernard Kolman 8/E

Module by: Julian Daza. E-mail the author

Summary: Algunos ejercicios resueltos sobre algebra lineal de Beernard Kolman

kol2005pgn245ejc21.tm por Julian Camilo Daza

21. Find de cosine of the angle betwen each pair of vectors u and v .

(a) u=(2,3,1)v=(3,-2,0)u=(2,3,1)v=(3,-2,0)

cos θ = | u v u v | cos θ = | u v u v |
(1)

sage: u=vector([2,3,1])

sage: v=vector([3,-2,0])

sage: u.dot_product(v)

0

sage: u.dot_product(u)

14

sage: v.dot_product(v)

13

cos θ = | 0 u v | cos θ = | 0 u v |
(2)

El angulo es 0.

(b) u=(1,2,-1,3)v=(0,0,-1,-2)u=(1,2,-1,3)v=(0,0,-1,-2)

sage: u=vector([1,2,-1,3])

sage: v=vector([0,0,-1,-2])

sage: u.dot_product(v)

-5

sage: u.dot_product(u)

15

sage: v.dot_product(v)

5

cosθ=|-553|cosθ=|-553|=|-13||-13|

θ = cos - 1 | - 1 3 | = 125 , 26 θ = cos - 1 | - 1 3 | = 125 , 26

(c) u=(2,0,1)v=(2,2,0)u=(2,0,1)v=(2,2,0)

sage: u=vector([2,0,1])

sage: v=vector([2,2,-1])

sage: u.dot_product(v)

3

sage: u.dot_product(u)

5

sage: v.dot_product(v)

9

cosθ=|335|cosθ=|335|=|15||15|

θ = cos - 1 | 1 5 | = 63 . 43 θ = cos - 1 | 1 5 | = 63 . 43

(d) u=(0,4,2,3)v=(0,-1,2,0)u=(0,4,2,3)v=(0,-1,2,0)

sage: u=vector([0,4,2,3])

sage: v=vector([0,-1,2,0])

sage: u.dot_product(v)

0

sage: u.dot_product(u)

29

sage: v.dot_product(v)

5

cos θ = | 0 u v | cos θ = | 0 u v |
(3)

El angulo es 0

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

### Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

### Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

### Reuse / Edit:

Reuse or edit module (?)

#### Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

#### Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.