Talk:Dot product/Archive 2

Archive 1

Archive 2

ommision in the image (?)

In the image the dotted line is at right angles to the vector B isn't it? Shouldn't the image indicate this? stib (talk) 01:55, 19 February 2008 (UTC)

I agree, the right angle sign should be appended to the image. I just learned about Dot Products, and that would have saved me a couple HOURS(!!) of reading. I didn't understand the image at all for a few days and that sign would have completely solved it for me. Now, granted, I'm rather ignorant of most maths, but surely this page isn't only for the highly literate mathematicians. If I should add the bracket myself, let me know, and let me know how. Thanks. Limited_Atonement

I edited the image to include the parallel indicators and |A||B|, but I don't know how to replace the existing image. I don't think I can upload to Wikipedia because my account isn't confirmed? How do I change that? Limited_Atonement

I successfully changed the image to indicate the right angle and took out the extra in |A||B|. Limited_Atonement Thursday, December 09, 2010 6:05:45 PM —Preceding unsigned comment added by 66.244.68.201 (talk) 18:06, 9 December 2010 (UTC)

Clarifiction on the dot product example

I have a small clarification to make in the article on Dot Product
(Mathematics).It is shown that dot product is matrix multiplication of a transpose multiplied by b
But in the example the transpose of b is taken.

Can you please calrify this 195.229.236.247 (talk) 09:04, 17 March 2008 (UTC) Aravind

Vector vs row or column vector

I don't think we need to distinguish between row and column vectors here. That distinction only needs to be made when considering vectors as degenerate forms of matrices. If vectors were always considered as degenerate matrices, we'd simply do matrix multiplication and forget about defining it as a vector operation.

Instead, we have separate operators (the dot or the Cartesian x) so that the vectors can remain vectors (as in vector-space), not specifically row or column vectors. The result is equivalent to considering the first vector as a row-vector and the second as a column-vector and applying normal matrix multiplication, but matrix multiplication is not commutative.

I believe the generalization (neither considered as row nor as column) makes dot product truly commutative. (additional operations such as transposition should not be artificially required by forcing row or column orientation on your model).

_-T 德 —Preceding unsigned comment added by 129.115.13.107 (talk) 16:29, 21 March 2008 (UTC)

Ambiguity in the introduction-definition

Is the dot product valid for "non-orthonormal vector spaces"? For readers who don't know the exact definition of the Euclidean space (i.e. most readers, in my opinion), the introduction does not answer this question explicitly. It just states that the dot product "is the standard inner product of the Euclidean space". Readers appreciate a concise and generic introduction, but they are not supposed to know whether the Euclidean space is required to be orthonormal or not. The definition section creates or accentuates this doubt. Here is the first sentence of that section:

The dot product of two vectors (from an orthonormal vector space) a = [a1, a2, … , an] and b = [b1, b2, … , bn] is by definition:

The phrase within parentheses puzzles me and may also confuse other readers. Why did the author use parentheses?

First hypothesis. Did the author mean that the dot product is only defined in orthonormal vector spaces? (and only its generalization, the inner product, can be used for non-orthonormal vector spaces)? In this case, the parenheses might indicate that the writer regards the information as redundant. However, this is not true for all readers, because the information is not given explicitly in the previous paragraphs.

Opposite hypothesis. Did the author mean, on the contrary, that a more general and complex definition of the dot product exists (coinciding with the definition of the inner product given below), valid for non-orthonormal vector spaces? Did the author choose to give first a commonly used simplified version of that definition? In this case, the definition section should be completed with the warning that a more general definition exists in the literature, and it coincides with the definition of the inner product.

What is the correct hypothesis? Do you agree that the phrase is an important part of the definition and should be enphasized, rather than confined within parenthesis? Paolo.dL (talk) 12:57, 19 April 2008 (UTC)

Here's what I think: the parenthesis was unhelpful (I've removed it). I think the dot product is naturally defined just using the formula that's given, and so we don't require an assumption that the basis of the vector space is orthonormal. I suppose if you took the geometric interpretation as the definition, then you'd need this qualification, but that's not the way the article is written, so the qualification is just noise here. That's my opinion! Ezrakilty (talk) 18:37, 12 May 2008 (UTC)

I believe that it is important to warn the reader that the definition of the dot product is not valid for non-orthogonal Euclidean vector spaces. Not valid means that it does not compute what is meant to compute (a number which can be used to define length and angle). I agree with Ezrakilty that this "validity" is related to the geometrical interpretation of the dot product, but IMO the geometrical interpretation is inseparable from the mathematical definition

$\mathbf {a} \cdot \mathbf {b} =\sum {a_{i}{\overline {b_{i}}}}$.

Moreover, if we accept Ezrakilty's decision to remove from the definition section the sentence about restriction to orthogonal vector spaces, we should remove it everywhere in the article, including the beginning of section "Conversion to matrix multiplication". Also, notice that at the end of the section "Generalization", the inner product formula is shown to simplify into the dot product formula when the basis set is orthogonal. This information is precious in this context.

Being not sure about what's the most common or most appropriate definition of the dot product, I undid Ezrakilty's edit, just to restore consistency between sections. I hope that the discussion on this topic will continue. Paolo.dL (talk) 14:44, 13 May 2008 (UTC)

Hi Paolo: I hear what you're saying. You're right to strive for precision, of course. On the other hand, I feel that this article presently mixes the narrow definition (the right one for many readers) with the more precise and general one (right for other readers). The opening is a case in point: it describes the dot product as an operation on "vectors over R" (a sensible statement for the narrow interpretation) but then takes a different tone, calling it "the standard inner product of the orthonormal Euclidean space." (Is that description even meaningful if we take vectors as simply tuples of reals?) I'd like to see the article separate the two viewpoints: it should first treat the dot product in the narrow sense and generalize only later. Then casual readers won't be scared off by the more abstract treatment. Ezrakilty (talk) 14:01, 15 May 2008 (UTC)

Your point is reasonable. Indeed, in physics textbooks, where I learned the definition of the dot product, the operation is described in R³ and is not even extended to n-dimensional spaces.

I guess that you accept my "first hypothesis". I still have a doubt. Can we exclude that some authoritative author might have endorsed, in the literature, my "opposite hypothesis" (see previous section)? In other words, does any mathematician call dot product the inner product for non-orthogonal vector spaces? I hope not. And I hope that some expert mathematician will give a final answer to this question either by posting a comment here or by editing the article. Paolo.dL (talk) 15:35, 15 May 2008 (UTC)

Removed ambiguous sentence

We still don't have an answer, but in the meantime I removed the ambiguous sentence. The last sentence of the definition section presents non-ambiguously a similar concept, hedged by the adverb "typically". Even if my "opposite hypothesis" were true (and I hope it's not), the ambiguous sentence would not be a good way to inform the readers. Paolo.dL (talk) 09:00, 30 May 2008 (UTC)

Decomposition and rotation

In the section "Properties," there's a spurious comment: "Decomposing vectors is often useful for conveniently adding them, e.g. in the calculation of net force in mechanics." Decomposition of vectors isn't defined anywhere in the article; but this is an interesting topic and perhaps it deserves treatment. Would anyone like to move this line to a new section and expand on the idea? Ezrakilty (talk) 18:46, 12 May 2008 (UTC)

People are supposed to know vectors before studying vector multiplications, such as the dot product. Decomposition is just how the scalar components (or direction cosines, or coordinates) of a vector are determined. It is useful for millions of reasons, not just for adding. And the topic is discussed in detail elsewhere. An enlightening example in this context is its application to vector or basis rotation. See if you like my edits. I moved in a separate subsection of "Geometric interpretation" the existing text about scalar projection, and created a new subsection called "Rotation". Paolo.dL (talk) 21:14, 1 June 2008 (UTC)

Please separate Maths from Physics

Quotting: "I think there could be two articles, the inner product space one giving the abstract algebraic formulation, and a more concrete geometric one focusing just on R^n or C^n, aimed more at people who might just use it for calculus or physics classes."

I STRONG agree with the separation proposed. PLEASE, do it (and a lot of students will be gracefull).

Armando Simoes, student —Preceding unsigned comment added by 83.132.150.145 (talk) 16:37, 26 November 2008 (UTC)

Conversion to matrix multiplication sections

I was wondering if the section dealing with the dot product as matrix multiplication should be extended to also cover the complex case in terms of the transpose conjugate:

$\mathbf {a} \cdot \mathbf {b} =\mathbf {b} ^{*}\mathbf {a}$

where

$\mathbf {b} ^{*}={\begin{bmatrix}{\overline {b}}_{1}&{\overline {b}}_{2}&\cdots &{\overline {b}}_{n}\end{bmatrix}}$

is the transpose conjugate. The ugly thing about this matrix notation is that the order of the vectors is reversed. It is therefore something you have to be careful about and which I think could be highlighted explicitly

The present real number formula

$\mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{T}\mathbf {b}$

could tempt casual readers looking for the complex relation (which they wont find) into generalizing this as $\mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{*}\mathbf {b}$ for the complex case, which is wrong.

Perhaps, if the real number example was extended to

$\mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{T}\mathbf {b} =\mathbf {b} ^{T}\mathbf {a}$

the casual reader looking for the complex version would think twice before generalizing by seeing the symmetry in the two equivalent expressions.

Actually, I would have personally preferred if the whole article had used the generalized complex version for defining the dot product and then treated the real case as a (very important) special case. I do see the problems in this as it may be a turn-off for the lay person reader, who may be intimidated by the notion of complex numbers. An alternative idea could be to completely split up the article into two articles. This one dealing exclusively with the dot product of real numbers and another article, e.g., dot product of complex vectors dealing with the generalized case. There should then be a reference to the generalized case from the preamble of this article.

Alternatively, the more abstract inner product article could be extended with a more thorough treatment of the dot product of complex vectors, including the matrix notation.

--Slaunger (talk) 10:25, 11 February 2009 (UTC)

I think it is important to have a thread running through the article that uses just real numbers, so that readers unfamiliar with complex numbers can still appreciate the dot product. Also, I think it's good to keep the more general definitions within this article, so that casual readers can have their interest piqued and perhaps extend their grasp.

All that said, when we can make formulas for the real case look like their complex friends, we should do so, as with your example

$\mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{T}\mathbf {b} =\mathbf {b} ^{T}\mathbf {a} .$

In this case, I think, we are not adding any burden for readers oblivious of complex numbers, yet as you note the temptation to generalize wrongly would be reduced.

--Ezrakilty (talk) 12:52, 11 February 2009 (UTC)

OK, then I suggest that the complex generalization

$\mathbf {a} \cdot \mathbf {b} =\mathbf {b} ^{*}\mathbf {a}$

is added to the matrix multiplication section and that the analogous symmetric notation of the dot product of real vectors in matrix notation is extended as I suggest. I would rather not do it myself though. I am not a native writer and I do not feel comfortable editing directly in an article here at en.

--Slaunger (talk) 18:55, 11 February 2009 (UTC)

A pedantic additional note; formally, it is not really correct to equate the two is it? the dot product is a complex scalar, whereas the $\mathbf {b} ^{*}\mathbf {a}$ is really a $1\times 1$ complex matrix. So I guess, that if the matrix multiplication version of the dot product should be also formally correct, it should be written as either

$\left[\mathbf {a} \cdot \mathbf {b} \right]=\mathbf {b} ^{*}\mathbf {a}$

or

$\mathbf {a} \cdot \mathbf {b} =\left(\mathbf {b} ^{*}\mathbf {a} \right)_{1,1}$

or is this so self-evident that it is not worth introducing the notational complexity?

I came to think of that after toying around with the dot product in an IPython shell using SciPy, where the distinction is very important to keep in mind.

In [1]: a = array([1, 1j]) # Vector notation
In [2]: b = array([1, -1j]) # Also as a vector
In [3]: a_dot_b = dot(a, b.conjugate()) # Explicit conjugation needed in SciPy
In [4]: print a_dot_b
0j
In [5]: a = mat('[1; 1j]') # As column matrix
In [6]: b = mat('[1; -1j]') # Also as columns matrix
In [7]: b_star_a = b.H * a
In [8]: print b_star_a
[[ 0.+0.j]]
In [9]: a_dot_b = b_star_a[0,0] # Pick the element from the matrix
In [10]: print a_dot_b
0j

--Slaunger (talk) 19:14, 11 February 2009 (UTC)

I think that in mathematics of this kind it is common to conflate such things as a scalar and a 1×1 matrix. The idea that a definition "is equivalent to" some other calculation often means just that you can arrive at the answer using the latter and little else. In computer science, however, and in some branches of mathematics, the "typing discipline" may be important--that is, a scalar and a 1×1 matrix are distinct and further computation is needed to transform between them. Even in category theory, where the typing discipline is usually important, it is common to seek values "up to isomorphism" and the isomorphism between scalars and 1×1 matrices justifies this.

— Ezrakilty (talk) 17:57, 12 February 2009 (UTC)

Concerning my latter pedantics your comment seems to make sense. It is not worthwhile introducing the notational complexity. But how about my original observation. Will you extend it along the lines of our discussions? -- Slaunger (talk) 18:32, 12 February 2009 (UTC)

<Math> Image not appearing in /*Conversion to matrix multiplication*/

Perhaps it is just me, but despite refreshing, the image described below does not appear. Instead, it's alt-text is shown. I am not familiar with Latex or whatever is used to generate these images, so I apologize that I cannot fix it myself.

\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^T \mathbf{b} \,

76.236.188.70 (talk) 02:42, 21 February 2009 (UTC)

Distinction between multiplication of two vectors and between a covector and a vector

In "Quick Introduction to Tensor Analysis" R.A. Sharipov calls a covector times a vector a scalar product and distinguishes it from a vector times a vector which he calls a dot product. This distinction isn't referenced in this article but perhaps it should be.

projection of a vector

When describing the projection of a vector. The text was confusing due to the close proximity of two separate definitions.

f neither a nor b is a unit vector, then the magnitude of the projection of a in the direction of b, for example, would be a · (b / |b|). b / |b| is the unit vector in the direction of b.

Changed this to hopefully remove ambiguity. —Preceding unsigned comment added by Fflanner (talk • contribs) 00:37, 10 March 2009 (UTC)

Wrong property

The fourth property

$(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} =(\mathbf {c} \mathbf {b} ^{T})\mathbf {a} .$

does not make any sense to me. This is clearly false when a and c are not parallel. --Marozols (talk) 15:06, 5 July 2009 (UTC)

Use of caps in "Geometric Interpretation" figure?

Is this article trying to maintain some consistency in the use of bold and capital versions of letters? Especially given that vectors and matrices are under discussion, this seems important... yet in the section "Geometric Interpretation" the narrative uses bold smalls for what appear to be the same vectors as the bold caps in the figure. Is this intentional or a mistake, or regarded as not significant? Gwideman (talk) 20:30, 18 March 2010 (UTC)

"cosine of the angle is returned"

In the Geometric Interpretation section, after the equation "theta = arccos(...)", appears the sentence "The cosine of the angle is returned". Clearly this does not refer to what the preceding equation returns, as this returns an angle. Probably the writer intended to talk about the meaning of the argument to arccos. Gwideman (talk) 20:29, 8 April 2010 (UTC)

Wrong and badly formatted caption

$A_{B}=|\mathbf {A} |\cos(\theta )$ is the scalar projection of A onto B.
Since A • B = $|\mathbf {A} ||\mathbf {B} |\cos(\theta )$, then A_B = (A • B) / $\mathbf {B}$.

This caption is badly formatted and contains a clearly wrong equation:

A_B = (A • B) / $\mathbf {B}$,

Parts of the equations are formatted with <math>, other parts are not. Moreover, in Wikipedia image captions are always rendered with a smaller font size, with respect to the text, so the <math> format is inappropriate. The previous caption was:

1. Completely correct
2. Consistently formatted
3. Did not use the excessively large <math> format

This is why I am going to revert this edit. If you do not agree, please first explain here the reason. Paolo.dL (talk) 21:22, 9 December 2010 (UTC)

Okay, thanks for starting a talk about this. I see that I missed an absolute value sign there and that four nowiki tags is way shorter than two math tag now.

Another question, what is one versus two pipes used to signify in this article? The one pipe notation has not been introduced at that point in the article. 0¹⁸ (talk) 00:37, 10 December 2010 (UTC)

See Table of mathematical symbols. In my opinion, two pipes are used when you are afraid that a single pipe can be confused with I. Sincerely, I have rarely seen university teachers using double pipe when they write on the board. The single pipe, which is the same symbol used for the absolute value of a scalar, but there's no ambiguity when it is applied to a vector, as nobody needs to take the absolute value of all the elements of a vector (it makes no sense).

And on the other hand, there's no ambiguity when you use it with a scalar, because the square root of the square of a scalar coincides with its absolute value. I mean, to compute the absolute value of a scalar you can take its Euclidean norm, if you like!

The problem is that in this article we use both conventions, and this is likely to confuse the reader. Another problem is that the image uses the single pipe. So, we can't use the double pipe in the caption. Feel free to change everything to single pipe. Paolo.dL

However, this is not the only difference between the figure and the text. Namely, the figure uses A and B, the text uses a and b It would be nice to have a picture using labels consistent with the text. Not vice versa, as vectors are typically represented by lowercase characters. (talk) 09:10, 10 December 2010 (UTC)

Actually, the MOS says a single pipe can be used for, "Euclidean distance" which is not as general as the double pipe, "norm of; length of". But a norm based on the dot product is going to be a 2-norm, and so a Euclidean distance. This is not the case for an inner product, but this page regards dot products.

I've always found the single pipe notation confusing when it doesn't mean absolute value or the argument isn't a complex number. But then others might find the norm notation confusing because it might imply a more complex norm than the 2-norm. For now, I've simply noted that it can be written either way. 0¹⁸ (talk) 15:25, 10 December 2010 (UTC)

The Table of mathematical symbols is clearly wrong. The single pipe does not denote "Euclidean distance". Otherwise, it would be written |a, b|. It denotes the length of a vector, as the distance is computed from 2 vectors, while the pipes act on one vector (which may or may not be a distance, i.e. the result of a difference). Paolo.dL (talk) 16:16, 10 December 2010 (UTC)

You might want to take this up there if you think you are right, but if you think of a vector connecting two points some distance apart, it makes perfect sense that |a| measures this distance. As for this article, do you care which is used? I do think consistency would make sense and the inner product section only works with two pipes. 0¹⁸ (talk) 17:10, 10 December 2010 (UTC)

Yes, but the pipes do not compute the difference. They just compute the length of that difference. The difference is represented by the minus sign, not by the pipes. In other words, the pipes are not equivalent to d(), otherwise (I repeat) Euclidean distance would be written |a, b| = d(a, b). Please see Euclidean distance. Two pipes are ok. Paolo.dL (talk) 18:31, 10 December 2010 (UTC)

Look at the second equation, the example is how a dot product is the E distance. 0¹⁸ (talk) 22:43, 10 December 2010 (UTC)

No, the square root of the dot product represents the Euclidean norm. The Euclidean norm of a difference is something more than a Euclidean norm. In other words, the Euclidean norm does not include the difference. In other words, The Euclidean distance is a sequence of operations: first you take the difference, then the Euclidean norm. Listen, I have repeated this concept three times. If you don't agree, I am sorry, but I won't change my mind about this, as this is elementary logic, not a subjective opinion. Paolo.dL (talk) 12:56, 11 December 2010 (UTC)

Does this impact the page? I understand what you are saying, there is no reason to repeat it. What if I defined the Euclidean distance as d(a,0) where 0 is the zero vector. Then I could write |a-0| = |a| and call it the Euclidean distance of a, no need for the second vector to be specified, since it is zero. This appears to be the leap that the authors of the style manual made and you are not willing to. 0¹⁸ (talk) 01:07, 12 December 2010 (UTC)

This is the "leap" that possibly led to their naive mistake. The concept of distance involves a difference between any two position vectors, so it requires a subtraction. I am sick of this nonsense. This is my last contribution to this discussion. Paolo.dL (talk) 11:18, 12 December 2010 (UTC)