Learning vectors: the dot product of vectors

In summary, The conversation is about learning dot products of vectors and how to approach certain expressions involving dot products. The main confusion is whether dot product symbol can be used to express multiplication between a scalar and a vector or if it is strictly for two vectors in the same vector space. The book being referenced in the conversation is attempting to clarify this concept but has not provided much explanation on the use of the dot product symbol.
  • #1
singleton
121
0
Well, I've been attempting to learn dot products of vectors and in doing so have come upon some questions.

NOTE:
a, b, and c are to be regarded as vectors
Please regard '*' as the dot for multiplication, too.

Question 1:
a * b * c

When I see this, I am unaware how to approach it. Should I tackle it left to right as I would normally?

a * b will yield a scalar value using the dot product. Then if I take the result and * c, I will have a scalar multiple times a vector. So the result is a vector?

OR do I do a1*b1*c1 + a2*b2*c2 + a3*b3*c3 and the result is a scalar?


Question 2:
(a * b) * c
I would naturally attack the brackets first of all and the resulting value will be scalar. Then continuing on, I have a scalar result value times the vector c. So, the final result is a scalar multiple of vector c, correct?
 
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  • #2
a.b.c makes no sense (a.b)c does as does a(b.c) and they are different. where does thsi come from? i mean who is using the same symbol for dot product (defined only on two vectors in the same vector space) and scalar mult?
 
  • #3
They are from questions in the book I'm attempting to learn from.

So, the dot product symbol cannot be used to express multiplication between scalar and a vector?

(I am unaware, I had always thought the dot was general multiplication, interchangable with the "x" symbol).

The book asks you to categorize the results of the expressions.

One of the categories is undefined, so that is where a.b.c would end up.
 
  • #4
yes the book is attempting to make you realize tht you can only form the dot product between two vectors (in the same space, so you can't dot a vector in R^2 with one in R^3)
 
  • #5
Once i saw a*b*c [or (a,b,c)], and it was supposed to be mixed multiplication [tex](\vec{a}\times\vec{b})*\vec{c}[/tex] (or [tex](\vec{a}*(\vec{b}\times\vec{c})[/tex], no difference) By x i mean vectorial multiplication.
 
  • #6
Thanks. This cleared up the problems I was having :)

I had encountered the dot before and wrongly assumed that it meant general multiplication, just another symbol. I was wrong ;)

Unfortunately the book didn't give to much on the use of the symbol. It just introduced it as "here, this is the algebraic formula for dot products and this is the geometric formula". Then it asked me those questions. So I was confused to say the least...

thanks again!
 

1. What is a learning vector?

A learning vector is a mathematical representation of a set of data or information. It is typically represented as a list or array of numbers, with each number representing a specific feature or attribute of the data.

2. What is the dot product of vectors?

The dot product of vectors is a mathematical operation that combines two vectors to produce a scalar (single) value. It is calculated by multiplying the corresponding elements of the two vectors and then summing the results.

3. How is the dot product used in learning vectors?

The dot product is used in learning vectors to measure the similarity or correlation between two vectors. It can also be used to project one vector onto another, allowing for comparisons and analysis of the data represented by the vectors.

4. What is the significance of the dot product in machine learning?

In machine learning, the dot product is used to calculate the similarity between data points and to identify patterns or relationships in the data. It is a crucial tool for tasks such as classification, clustering, and dimensionality reduction.

5. How can understanding the dot product of vectors improve data analysis?

Understanding the dot product of vectors can improve data analysis by providing a way to measure the similarity between data points and to identify important features or relationships in the data. It also allows for more efficient and accurate calculations in machine learning algorithms.

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