Talk:Dot Product

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Checklist

My comments are in red AnnaP 7/7

The proof for property 5 still needs some work. There's a typo, and you need to make sure that you have a sentence showign the final equality, saying that, "So we have shown..." Also, instead of having "Go through the same process and get: " sound like a command, you can say "You can go through this same process to get: " AnnaP 7/12

Ljeanlo1 20:07, 13 July 2011 (UTC) Anna, I fixed the typo (I accidentally wrote an a where an s should have been). Also, I have added the phrases you provided.

Message to the Future

  • I don't have any suggestions that future generations might want to add.
  • Maybe, someone might want to tackle cross product to show other cool things that happen with vectors. I always learn these two topics as a pair.

Reference and Footnote

  • All images are my own.
  • Didn't use any direct quotes from textual sources are cited.
  • Included references at the end of the page

Quality of Writing

  • I think the writing is clear and concise. The purpose is defined in each section.
  • There aren't that many pictures. But the images are directed.
  • terms are defined and are in bold text
  • Proof is included and hidden in case it may seem overwhelming
  • Proof of properties are also given

  • Add some guiding text to your proofs, just to explain each step in words to the reader. This should happen throughout the page.

  • Rebecca 01:18, 8 July 2011 (UTC) I agree with Anna's comments that you should add transition sentences between steps of the proofs.

  • Ljeanlo1 15:46, 8 July 2011 (UTC)I added a description of the properties.

I just realized that all of your properties work in n dimensional space, but your proofs are only for 3D space. There should be some note of that somewhere. Also, I do still think that there should be text between the steps to explain what you are doing.

  • Ljeanlo1 19:35, 11 July 2011 (UTC) For some reason, I misunderstood you the first time. I tried to explain myself, but had trouble being concise. Take a look. As for the other comments I did those.

Also, on property 5, don't just use the word "similarly" You can see my note on Richard's Law of Sines discussion page for a longer explanation of why, but the short version is that "Similarly," isn't enough explanation for a reader.
Turn your expressions after this sentence 'We use the distributive property of the dot product. " into equations (eg, bring the  | \vec{C}|^2 down from the previous line
It would also be very helpful to give the two equations that you combine in that sections numbers. That way you can say "combining equations 1 and 2..." and the reader will know where the next lines come from.
  • I'd suggest finding another way to explain our motivation for using the angle formula instead of talking about a vector with 10000 components, particularly since you state later on that you can only use this interpretation in 3D space. I's suggest explaining it using a more geometrical argument.

  • Ljeanlo1 15:46, 8 July 2011 (UTC) I added a few sentences and an analogy. But, I'm still not sure if I addressed what you wanted me to address.

  • This sentence: "This follows the first definition we mapped onto the dot product. " is unclear. Can you explain it in a different way?

  • Ljeanlo1 15:46, 8 July 2011 (UTC) explained it in a different way

  • You use this sentence "Now that we know the basics of the Law of Cosine we will prove that the alternative definition of the dot product is correct. " to set up for your proof, but then also say "We will use the Law of Cosine to prove the dot product." just two sentences later. Pick one of the two--it's redundant as is.

  • Ljeanlo1 15:46, 8 July 2011 (UTC) I took out one of the sentences.

  • Instead of saying "(Refer to the Properties of Dot Product listed and proved below)" link to what you're referring to.

  • Ljeanlo1 15:46, 8 July 2011 (UTC) did this.

Mathematical Accuracy

  • Can you use either parentheses or normal brackets to denote vectors? Why I ask is that the angle bracket that you're using to denote vectors is often used to denote the inner product. It would clear things up a bit.
  • You don't need to put any restrictions on the angle in your geometrical explanation section. It's fine if you want to, but it's not totally necessary

  • Ljeanlo1 15:46, 8 July 2011 (UTC) did this

Examples

  • In your first example, can you add some guiding words? So instead of just saying "For example" and showing the calculation, can you say "For example, if A= and B=, then A dot B= That will make things more clear to the reader.
  • In your "real world example" one of your vectors needs a new name. They can't both be A.

  • Ljeanlo1 15:46, 8 July 2011 (UTC) did this.

Layout

  • page was viewed in different window sizes and looked good

*When you go through the proofs of the properties, put each property either in bold or as a subheading to make it jump out more

  • Make your image showing the projection a bit bigger
  • Also, make the image for your example of projection bigger. If possible, make the angle look more like a 40 degree angle on your picture (it looks more like a 60 or 70 degree angle as is)

  • Ljeanlo1 15:46, 8 July 2011 (UTC) did this.

  • Rebecca 01:16, 8 July 2011 (UTC) I would move the discussion of inner product to a later part of the page. It's clearly written, but it isn't necessary to understand the beginning of the page, and it complicates things a bit.

  • Rebecca 01:17, 8 July 2011 (UTC) It looks like there's something weird under some of the "click here to see more" links. Do you know why this is happening?

  • Ljeanlo1 15:46, 8 July 2011 (UTC) I like where the inner product is mentioned, because its a quick preview and something someone doesn't need to really focus on. Also, the click here to see more is working fine for me and the computers in the lab. but I'll check again.

General Comments

Kate 13:57, 9 June 2011 (UTC):

  • I mentioned this at the meeting yesterday, but there's already a Vector page. I think most of what you have under your vector section could go onto that page, especially considering that page has almost no unhidden text.
Leah 11:00, 13 June 2011 (UTC) Remodeled the vector page and took this information out.

  • Be careful with your notation throughout. There are lots of ways to write vectors, but I think you should pick one and stick with it. (You've used capital letters, lowercase letters, capital letters with arrows, AND lowercase letters with arrows.)
Leah 11:00, 13 June 2011 (UTC) Incorporated other notations, showed how things are denoted.

  • Read over your sentences. A couple of them are a little awkwardly phrased.

Leah 11:00, 13 June 2011 (UTC) could you point them out if they still sound iffy?

*Rebecca 16:57, 11 June 2011 (UTC) I think you have a really good start here! I like the way you introduce the topic especially, and I think your progression through topics is very logical.

xd 13:48, 22 June 2011 (UTC) In the opening paragraph, you mentioned and bolded "scalar product" and "inner product". So I am expecting definition and explanation for those guys. These things are really not the same thing. For example, inner product is the general term for a vector space. Prof. Maurer can give you a much better explanation for this.

Section-specific Comments

Vectors

  • Rebecca 22:22, 10 June 2011 (UTC) I think you should put the link to the vectors page in the first sentence and delete the section sentence of the very first section.
  • Rebecca 17:10, 11 June 2011 (UTC) Now that I'm thinking about it more, I think the first sentence of your basic description could replace your whole first section under "dot product". I think that description is a bit confusing, and I think this is a better into "The dot product is a way of multiplying two vectors to get a quantity with only magnitude. We use the dot product to compute lengths of vectors and angles between vectors." You can still link to the vectors page in the first sentence. You could add a last sentence that says "It is also called the scalar product or the inner product if you think that part should stay.
  • Leah 11:12, 13 June 2011 (UTC) Did Rebecca's suggestion.

Kate 13:57, 9 June 2011 (UTC):

  • There were a couple things about this section that confused me. The way you introduced i, j, and k didn't really make sense - just saying that they correspond to x, y, and z doesn't actually help me understand what they are. (I don't think you even mentioned that they all have length 1, which is super important!)
  • Leah 11:12, 13 June 2011 (UTC) moved into vector page.

*Related to that, I really think that you should at least point out that vectors can be written as tuples. I've always found (3, 3, 2) much easier to understand than 3i + 3j + 2k, and I think describing i, j, and k as tuples would help make them clearer.

  • Leah 11:12, 13 June 2011 (UTC) moved to vector page and offer different ways of writing vectors.
  • Also, it'd be really helpful if you drew up some 2-D things, like "this is what the vector (3,2) or 3i + 2j looks like"

*Lastly, you say "vectors are written with hats" and then later "vectors are written with arrows", which might seem contradictory

  • Leah 11:12, 13 June 2011 (UTC) moved all information and changes to vector page.

Basic Description

Kate 13:57, 9 June 2011 (UTC): The shopping example is unfamiliar to me, but I think it could work if you did a better job tying it to the actual math. Make it explicit which things are the vectors, what their components are, and what their dot product is. [CHECK]

I also think it would be helpful to compute a couple sample dot products in this section, and maybe even prove the part about how A dot B is ABcosθ. (If you do that proof, remember to add pictures!) [CHECK- used the law of cosine unsure if I should prove it another way]

* Rebecca 17:14, 11 June 2011 (UTC) I agree with Kate. The shopping example is a great idea, but it need to be explained more. I understand what you're saying, but the way it's worded makes it confusing I think. Here's my suggestion...

You say "The price of an item is one vector and the number of items is another vector. The component of the vector is represented by the number of different supplies being bought. So we would multiply the price of the pencils and the number being bought then we would add that product to the price of folders times the number of folders and we would keep going until we reach our last component."
I think this needs to be changed so that it is a bit more explicit. I think what you need to say is that "The prices of items are components of one vector, and the quantities you are buying of those items make up another vector. To calculate the total cost of your school supplies, you just take the dot product of these vectors."
Then I would give an example using real numbers. "For example, suppose you need to buy 3 pencils, 4 folders, and 5 notebooks. Pencils cost $2, folders cost $0.50, and notebooks cost $1.
The vector for quantities of the items is \mathrm{A} = (3,4,5), and the vector for prices of the items is \mathrm{B} = (2, .5, 1).
Taking the dot product of these vectors gives us
 (3, 4, 5) \cdot (2, .5, 1)
Using the formula for dot product, we have
 3 \times 2 + 4 \times .5 + 5 \times 1
Simplifying, we have

 6 + 2 + 5 = 13

  • In order to add this example though, you're going to need to move this section to the beginning of the basic description....

In the general case, the dot product is the summation of the products of the vectors' components. We have two vectors A and B. Both vectors have n components. The definition of the dot product states

Let \mathrm{A} = \left \langle a_1, a_2, \cdots, a_n \right \rangle. Let \mathrm{B} = \left \langle b_1, b_2, \cdots, b_n \right \rangle.
Then, \mathrm{A} \cdot \mathrm{B} = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n
  • So, to summarize, I would say you should revamp your basic description. I'd move the first paragraph to the previous section, I'd move the part about the general case before the shopping example, and then I'd add the numerical example to the shopping section.

[CHECK-DID ALL THE SUGGESTIONS. REORGANIZED IT AND ADDED THE EXAMPLE GIVEN. VERY HELPFUL !]

*xd 13:52, 22 June 2011 (UTC) Be aware that you are using different brackets and parenthesis for the vectors.

Properties of the Dot Product

Kate 13:57, 9 June 2011 (UTC): I think there could be a little bit more here than just a list of properties - why are these useful? Also, you might want to explain the length-squared notation, people might not be familiar with |A| meaning that. [WORKING ON THIS STILL. HOPING TO ADD A PARAGRAPH EXPLAINING PROPERTIES ETC. AND I INCLUDED THE MEANING OF |A| IN THE VECTOR PAGE]

Rebecca 22:24, 10 June 2011 (UTC) Good first paragraph of this section.

Applications

Kate 13:57, 9 June 2011 (UTC): I found this section really unclear. Why and how does the dot product do all of those things? Especially the perpendicular bit - you need to mention somewhere that that happens when it's zero. [TOOK AWAY THIS SECTION]

More Math Section

Alternative Definition

*xd 13:59, 22 June 2011 (UTC) Be aware of the link between your first general definition and the cosine definition. I believe that the cosine definition is only defined in the Euclidean space, that is 3 dimensional space. When you have n dimensional space, then your angle will have new definition which you have not supplied. So you have to very exact and precise when you offer new definition. In addition, you have to be aware of the consistency you represent your vectors. You either bold all of them or put a hat on all of them. You cannot do half and half. You can consult those who has taken math 28H or 28. I think there are plenty who have done that in the room.

  • xd 14:04, 22 June 2011 (UTC) \vec A ^2 and \vec B ^2 are not defined. You have to use \left | \vec A \right | ^2. Also, you might want to just mention that \left | \vec A \right | denotes the length of the vector.
  • "We will use the Law of Cosine to prove the dot product.
\left | \vec C \right | ^2 = \vec C \cdot \vec C = \left ( \vec A \cdot \vec B \right ) \cdot \left ( \vec A \cdot \vec B \right )"

xd 14:09, 22 June 2011 (UTC) I believe you mean A minus B right? In the next step, you used the distributive property of dot product. I think you should mention this property (I see you have the properties below so you can just refer to that) Alternatively, you can list the ten axioms of a vector space and justify from there.

Leah 22 June 2011 I'll make sure the ten axioms are mentioned in the vector helper page.

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