Search results “Vectors dot product”

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Vectors - The Dot Product. I show how to compute the dot product of two vectors, along with some useful theorems and results involving dot products. 3 complete examples are shown.

Views: 865874
patrickJMT

This physics & precalculus video tutorial explains how to find dot product of two vectors and how to find the angle between vectors. In addition, it explains how to determine if two vectors are orthogonal / perpendicular, parallel, or neither using the dot product formula and by analyzing the slope. This video contains plenty of examples and practice problems.
Here is a list of topics:
1. Dot Product of Two Vectors
2. How to Find the Angle Between Two Vectors Using Cosine & Tangent
3. How to find the magnitude of a vector
4. Geometric Interpretation of Vectors
5. Parallel and Perpendicular Vectors
6. How to determine if two vectors are orthogonal using the dot product formula and by finding the slope
7. How to express a vector in component form given magnitude and angle
8. Dot product of Two Vectors Using Three Components in i j k
9. Graphical Representation of Vectors

Views: 36454
The Organic Chemistry Tutor

Introduction to the vector dot product. Created by Sal Khan.
Watch the next lesson: https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/electric-motors/v/dot-vs-cross-product?utm_source=YT&utm_medium=Desc&utm_campaign=physics
Missed the previous lesson? https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/electric-motors/v/magnetism-11-electric-motors?utm_source=YT&utm_medium=Desc&utm_campaign=physics
Physics on Khan Academy: Physics is the study of the basic principles that govern the physical world around us. We'll start by looking at motion itself. Then, we'll learn about forces, momentum, energy, and other concepts in lots of different physical situations. To get the most out of physics, you'll need a solid understanding of algebra and a basic understanding of trigonometry.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
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Views: 440535
Khan Academy

We learned how to add and subtract vectors, and we learned how to multiply vectors by scalars, but how can we multiply two vectors together? There are two ways to do this, and one of them is called the dot product. Let's learn how to find the dot product of two vectors now!
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Professor Dave Explains

Dot products are a nice geometric tool for understanding projection. But now that we know about linear transformations, we can get a deeper feel for what's going on with the dot product, and the connection between its numerical computation and its geometric interpretation.
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3Blue1Brown

I created this video with the YouTube Video Editor (http://www.youtube.com/editor)

Views: 175834
cbseclass videos

Definitions of the vector dot product and vector length
Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/proving-vector-dot-product-properties?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra
Missed the previous lesson?
https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/subspace_basis/v/linear-algebra-basis-of-a-subspace?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra
Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
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Views: 650773
Khan Academy

Visual interpretation of the cross product and the dot product of two vectors.
My Patreon page: https://www.patreon.com/EugeneK

Views: 389143
Physics Videos by Eugene Khutoryansky

YOUTUBE CHANNEL at https://www.youtube.com/ExamSolutions
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Views: 151439
ExamSolutions

Visit http://ilectureonline.com for more math and science lectures!
In this video I will explain the products of vectors, or dot product, or scalar product.
Next video in this series can be seen at:
https://youtu.be/ffAzfVYgDII

Views: 67504
Michel van Biezen

Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Dot Product : Find Angle Between Two Vectors.
Here I do another quick example of using the dot product to find the angle between two vectors.

Views: 335882
patrickJMT

This physics video tutorial shows you how to add or subtract vectors to calculate the magnitude and direction / angle of the resultant force vector ad how to perform operations such as the dot product and cross product of two vectors. This video contains plenty of examples and practice problems.
Here is a list of topics covered in this video:
1. How To Add or Subtract Two Vectors Graphically
2. Head to Tail Method of Vector Addition & Subtraction
3. Adding & Subtracting Vectors Using The Parallelogram Law
4. How To Add Vectors Analytically or Algebraically
5. Parallel & Perpendicular Addition of Vectors - Right Triangle
6. How To Find The Resultant of Two or Three Vectors
7. How To Find The Magnitude & Direction of The Resultant Force
8. Reference Angle & The Vector Angle Measured From The Positive X Axis
9. How To Find The Components of a Vector - X and Y Component or Fx and Fy From F - Magnitude & Theta - Direction Angle
10. How To Represent a Vector in Cartesian Notation - ijk
11. How To Add or Subtract Two Vectors in Cartesian Notation
12. The Scalar Multiplication of Two Vectors - Dot Product
13. How To Find The Dot Product of Two Vectors in 2D and 3D
14. How To Find The Dot Product Using The Magnitude of Two Vectors and The Angle Between Them
15. How To Calculate The Magnitude of a vector
16. How To Find The Angle Between Two Vectors Using Inverse Tangent & The Dot Product
17. Work = Force x Displacement x cosine theta - an example of the dot product in real life physics: W=FDcosx
18. How To Tell If Two Vectors Will be Orthogonal or Perpendicular
19. How To Find The Angle Between Two 3 dimensional vectors Using The Dot Product
20. The Cross Product of Two Vectors To Produce a New Vector
21. Proving The Cross Multiplication of Two Vectors Creates a New Vector That's Perpendicular To Both Original Vectors
22. Cross Product Example: F = BqV sinx or Magnetic Force = Magnetic Field x Charge x Velocity x sine (theta)
23. Centripetal Force & Circular Motion of a Proton in a Magnetic Field
24. How To Find The Cross Product of Two Vectors Using Cartesian Vector Notation
25. How To Find The Cross Product Using The Magnitude of Two Vectors and The Angle Between Them
26. How To Find The Resultant Vector Using Vector Notation
27. How To Find The Resultant Force Vector Using The Law of Cosines
28. How To Find The Coordinate Direction Angles of a three dimensional Vector
29. Unit Vector Notation - How To Find The Unit Force Vector

Views: 165540
The Organic Chemistry Tutor

Understanding the differences between the dot and cross products. Created by Sal Khan.
Watch the next lesson: https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/electric-motors/v/calculating-dot-and-cross-products-with-unit-vector-notation?utm_source=YT&utm_medium=Desc&utm_campaign=physics
Missed the previous lesson? https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/electric-motors/v/the-dot-product?utm_source=YT&utm_medium=Desc&utm_campaign=physics
Physics on Khan Academy: Physics is the study of the basic principles that govern the physical world around us. We'll start by looking at motion itself. Then, we'll learn about forces, momentum, energy, and other concepts in lots of different physical situations. To get the most out of physics, you'll need a solid understanding of algebra and a basic understanding of trigonometry.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Physics channel: https://www.youtube.com/channel/UC0oGarQW2lE5PxhGoQAKV7Q?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy

Views: 424149
Khan Academy

Calculus 3 Lecture 11.3: Using the Dot Product: Explanation of the Dot Product, Finding the angle between two vectors including how the Dot Production show orthogonal vectors, Vector Projection, Work, and Direction angles.

Views: 117673
Professor Leonard

This is part of the Introduction to Electrodynamics series. Here, I compare the dot and cross products of two vectors from a geometric perspective.

Views: 198186
Real Physics

Donate here: http://www.aklectures.com/donate.php
Website video link: http://www.aklectures.com/lecture/dot-product-cross-product-and-multiplying-vectors-by-scalars
Facebook link: https://www.facebook.com/aklectures
Website link: http://www.aklectures.com

Views: 60865
AK LECTURES

This physics video tutorial shows you how to find the dot product of two vectors represented in component form. It provides the formulas and equations needed for the calculation as well providing plenty of examples and practice problems. This video also shows you how to find the angle between two vectors.

Views: 15310
The Organic Chemistry Tutor

For COMPLETE Vector Algebra please check this playlist -
https://www.youtube.com/watch?v=eUxRIHahFs0&list=PLuxAKwDAh21CLvNbdCsYZH5FzaoqbMQRn
OTHER TOPICS -
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https://www.youtube.com/watch?v=Rf9S858z41g&list=PLuxAKwDAh21BS3dPDHTtsm7A_bpttsAc8
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https://www.youtube.com/watch?v=wu70imcFElM&list=PLuxAKwDAh21DC9FB-VPSpT2GMSLMhVx-c
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https://www.youtube.com/watch?v=rG6PD5TJUKE&list=PLuxAKwDAh21DSO4rHYi7JNkpmZoyw-BX4
Probability
https://www.youtube.com/watch?v=toSW6WF_iaU&list=PLuxAKwDAh21CesK5RhiGdvZICRvAS7lml
Differentiation
https://www.youtube.com/watch?v=e9VGKUC5N88&list=PLuxAKwDAh21Dt4iFpTiVh5oZ7mC3ZUoDZ
Integration
https://www.youtube.com/watch?v=Znb5v2VfCxU&list=PLuxAKwDAh21At3EVYD_BAo_72ZcjpyRRg

Views: 124534
studyezee

The angle between two vectors can be found by using the cosine rule in a clever way.
Learn by viewing, master by doing
www.virtuallypassed.com

Views: 7674
Virtually Passed

What is Scalar Product & why it is defined the way it is defined

Views: 91969
The Lighthouse

This physics video tutorial explains how to find the cross product of two vectors using matrices and determinants and how to confirm your answer using the dot product formula. This video contains plenty of examples and practice problems using the components i j k.

Views: 51978
The Organic Chemistry Tutor

This video explains why the dot product is about how much vectors point the same way.

Views: 6837
Looking Glass Universe

Determine if two vectors are perpendicular by checking if the inner product (dot product) is equal to zero.

Views: 49511
Wuoti Math

Calculating the dot and cross products when vectors are presented in their x, y, and z (or i,j, and k) components. Created by Sal Khan.
Watch the next lesson: https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-flux-faradays-law/v/flux-and-magnetic-flux?utm_source=YT&utm_medium=Desc&utm_campaign=physics
Missed the previous lesson? https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/electric-motors/v/dot-vs-cross-product?utm_source=YT&utm_medium=Desc&utm_campaign=physics
Physics on Khan Academy: Physics is the study of the basic principles that govern the physical world around us. We'll start by looking at motion itself. Then, we'll learn about forces, momentum, energy, and other concepts in lots of different physical situations. To get the most out of physics, you'll need a solid understanding of algebra and a basic understanding of trigonometry.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Physics channel: https://www.youtube.com/channel/UC0oGarQW2lE5PxhGoQAKV7Q?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy

Views: 320025
Khan Academy

Proving the "associative", "distributive" and "commutative" properties for vector dot products.
Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/proof-of-the-cauchy-schwarz-inequality?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra
Missed the previous lesson?
https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/vector-dot-product-and-vector-length?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra
Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to KhanAcademy’s Linear Algebra channel:: https://www.youtube.com/channel/UCGYSKl6e3HM0PP7QR35Crug?sub_confirmation=1
Subscribe to KhanAcademy: https://www.youtube.com/subscription_center?add_user=khanacademy

Views: 306494
Khan Academy

Article: https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/

Views: 42025
Better Explained

This physics video tutorial shows you how to calculate the dot product and cross product of two vectors using cartesian vector notation. It also shows you how to calculate the dot product and cross product given the magnitudes of the two vectors and the angle between them. It also explains the cross product of the following: ixj=k, jxk=i, kxi=j, jxi=-k, kxj=-i, ixk=-j, ixi=0, jxj=0, kxk=0

Views: 11200
The Organic Chemistry Tutor

Mathematics, Class XII Chapter: products of vectors Topic: Scalar products Classroom lecture by Swati Mishra. Language : English mixed with Hindi.

Views: 28542
Pradeep Kshetrapal

Online Graphics Course: Math Review. First segment on vectors and dot products.
Table of Contents:
00:00 - Foundations of Computer Graphics
01:52 - Motivation and Outline
02:53 - Vectors
04:11 - Vector Addition
05:14 - Cartesian Coordinates
07:18 - Dot (scalar) product
08:19 - Dot (scalar) product
09:22 - Dot product in Cartesian components
09:37 - Dot product in Cartesian components
11:35 - Dot product: some applications in CG
12:06 - Projections (of b on a)

Views: 42088
raviramamoorthi

I explain how to find the Dot Product and the properties of the dot product. I continue by explaining how to calculate the angle between to vectors, including the special cases of Parallel Vectors and Orthogonal Vectors.
Check out http://www.ProfRobBob.com, there you will find my lessons organized by class/subject and then by topics within each class. Find free review test, useful notes and more at http://www.mathplane.com

Views: 61957
ProfRobBob

The super easy and quick concept of Dot product of vectors or dot multiplication of vectors or scalar multiplication of vectors is explained in this video.
The video covers all about the dot product and scalar produc of vectors. I will also tell you about:
1) Dot Product and Scalar Product of Vectors, Physics.
2) Physics - Mechanics: Vectors (12 of 21) Product Of Vectors: Dot Product
3) The Dot Product.
4) Class 12 Maths : Vectors - Dot product ( Scalar product ) Part 1.
5) Scalar Product or Dot Product (Hindi).
6) Cross Product and Dot Product: Visual explanation.
7) Dot vs. cross product | Physics | Khan Academy
Dot product of vectors or scalar product of vectors is the multiplication of two or more vectors. Dot product or scalar product of two vectors always give us a number.
a scalar function of two vectors, equal to the product of their magnitudes and the cosine of the angle between them.
The Dot Product gives a number as an answer (a "scalar", not a vector).
The Dot Product is written using a central dot:
a · b
This means the Dot Product of a and b
We can calculate the Dot Product of two vectors this way:
dot product magnitudes and angle
a · b = |a| × |b| × cos(θ)
Where:
|a| is the magnitude (length) of vector a
|b| is the magnitude (length) of vector b
θ is the angle between a and b
he dot product between two vectors is based on the projection of one vector onto another. Let's imagine we have two vectors aa and bb, and we want to calculate how much of aa is pointing in the same direction as the vector bb. We want a quantity that would be positive if the two vectors are pointing in similar directions, zero if they are perpendicular, and negative if the two vectors are pointing in nearly opposite directions. We will define the dot product between the vectors to capture these quantities.
But first, notice that the question “how much of aa is pointing in the same direction as the vector bb” does not have anything to do with the magnitude (or length) of bb; it is based only on its direction. (Recall that a vector has a magnitude and a direction.) The answer to this question should not depend on the magnitude of bb, only its direction. To sidestep any confusion caused by the magnitude of bb, let's scale the vector so that it has length one. In other words, let's replace bb with the unit vector that points in the same direction as bb. We'll call this vector uu, which is defined by
u=b∥b∥.
u=b∥b∥.
The dot product of aa with unit vector uu, denoted a⋅ua⋅u, is defined to be the projection of aa in the direction of uu, or the amount that aa is pointing in the same direction as unit vector uu. Let's assume for a moment that aa and uu are pointing in similar directions. Then, you can imagine a⋅ua⋅u as the length of the shadow of aa onto uu if their tails were together and the sun was shining from a direction perpendicular to uu. By forming a right triangle with aa and this shadow, you can use geometry to calculate that
a⋅u=∥a∥cosθ,(1)
(1)a⋅u=∥a∥cosθ,
where θθ is the angle between aa and uu.
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Views: 4814
Najam Academy

Thanks for appreciation
You can also find the perpendicular vectors by Cross Product.
Also watch that Playlist Vectors: Cross Product

Views: 9449
Anil Kumar

In this video I have explained how to find scalar and vector product of two vectors , how to find scalar and vector product of two vectors when both the vectors are given in component forms .
The dot (scalar) product is the trigonometric cosine rule applied to vectors. The result of a dot product is the cosine of the angle between the two vectors multiplied by the magnitudes of the vectors.
The cross product is a vector at right angles to both of the given vectors and whose length is equal to the sine of the angle between the vectors multiplied by the magnitudes of the vectors. The length of the cross-product vector is also equal to the area of the parallelogram formed by these vectors.
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Dhiman Rajesh Dhiman

Subscribe! http://www.freemathvideos.com Want more math video lessons? Visit my website to view all of my math videos organized by course, chapter and section. The purpose of posting my free video tutorials is to not only help students but allow teachers the resources to flip their classrooms and allow more time for teaching within the classroom. Please feel free to share my resources with those in need or let me know if you need any additional help with your math studies. I am a true educator and here to help you out.

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Brian McLogan

Vectors - Dot Products - Cross Products - 3D Kinematics - Great Demos
Assignments Lecture 1, 2, 3 and 4: http://freepdfhosting.com/614a811c6d.pdf
Solutions Lecture 1, 2, 3 and 4: http://freepdfhosting.com/90c2930d13.pdf

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Lectures by Walter Lewin. They will make you ♥ Physics.

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This Video Lecture Series presented By VEDAM Institute of Mathematics is Useful to all students of Class 11th & 12th of CBSE, ICSE, MP Board and other boards of India.
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Jaipal Vishwakarma

Visit http://ilectureonline.com for more math and science lectures!
In this video I will show you how to calculate the dot product of 2 vectors, example 1.
Next video in this series can be seen at:
https://youtu.be/N3Yy5EvLRbc

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Michel van Biezen

YOUTUBE CHANNEL at https://www.youtube.com/ExamSolutions
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ExamSolutions

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Shridhar Jagtap

In this short tutorial we will learn how to use the vector functions on the TI-36X Pro. This will allow us to manipulate vectors and find dot and cross products.

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Calculator Expert

This video provides an overview of the formula for and applications of the vector dot product. It covers scalar projection. It serves as the basis for the next video about Path Following.
Read along: http://natureofcode.com/book/chapter-6-autonomous-agents/#chapter06_section7
More info: http://en.wikipedia.org/wiki/Dot_product
Code from video:
https://github.com/shiffman/The-Nature-of-Code-Examples/tree/master/chp06_agents/Exercise_6_09_AngleBetween
https://github.com/shiffman/The-Nature-of-Code-Examples/tree/master/chp06_agents/SimpleScalarProjection
Help us caption & translate this video!
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The Coding Train

For Guidance Contact : [email protected]

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Anil Kumar

This concept is explained concisely by IITian Faculty.
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Uniclass Content

Dot and Cross Product Comparison/Intuition
Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/vector-triple-product-expansion-very-optional?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra
Missed the previous lesson?
https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/proof-relationship-between-cross-product-and-sin-of-angle?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra
Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
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Khan Academy

https://bit.ly/PG_Patreon - Help me make these videos by supporting me on Patreon!
https://lem.ma/LA - Linear Algebra on Lemma
https://lem.ma/prep - Complete SAT Math Prep
http://bit.ly/ITCYTNew - My Tensor Calculus Textbook

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MathTheBeautiful

Using Dot Product to Find the Angle Between Two Vectors. You can use one of the dot product formulas to actually compute the angle between two vectors. Here we show how to use this formula to find an angle theta between two vectors Subscribe on YouTube: http://bit.ly/1bB9ILD
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© 2019 Making funny faces with food

Under the late republic, the number of provinces rapidly increased, and therefore, Pompey the Great proposed a new law, the Lex Pompeia de provinciis , in which former praetors and consuls were obliged to become governor five years after their term in office (53). At more or less the same time, he had himself elected as governor of several provinces, which were not governed by himself, but by his representatives, the legati . The emperor Augustus copied this idea when he changed the empire, until then ruled as a republic, into a monarchy. He was made governor of almost all provinces with legions, and used legati to rule them. At the same time, the rest of the empire was governed by proconsuls. So, there were two types of governors: Proconsuls. In fact, these men were not former consuls, but former praetors. They governed the senatorial provinces and typically served twelve months. Only the rich provinces -Asia and Africa- were entitled to a proconsul who was indeed an ex-consul. Legati Augusti pro praetore. These men served in the emperors provinces with the armies (the imperial provinces ). Usually, their term in office lasted thirty-six months, although the emperor Tiberius preferred longer terms. There was a third group of governors. In several unimportant provinces, prefects were appointed. Usually, these military men governed parts of larger provinces. The best known example is Pontius Pilate, who governed Judaea, an annex to Syria. Prefects were not senators but knights. Egypt was also governed by a prefect, not because it was unimportant, but because it was the emperors own possession. When Septimius Severus conquered Mesopotamia, he used the same construction. After the mid-first century, the prefects were gradually replaced by procurators (except for Egypt). The only difference is that prefects were soldiers and procurators were fiscal officials. It tells something about the success of the Pax Romana .