Re: Dot Product

[luffy]

Hey guys, I have a very simple theory question,

What is the Dot Product? I don't mean "how do u find the dot product?" I mean, when you are finding the dot product of lets say vector 'a' and vector 'b'. What are you actually doing?

Many people have told me it is the "magnitude of a in the direction of b". However, I don't see how this makes sense, because if this was the case, then a.b = a.b(hat), which obviously is not true.

I just wanted to know, so that I will have a better understanding of 'applying' vectors.

[samiira]

Geometrically, the dot product (aka scalar product) of two vectors is the magnitude of one times the projection of the other along the first

Let's say we have two vectors, A and B.  The values |A| and |B| represent the lengths of vectors A and B, respectively, and Θ is the angle between the two vectors. The dot product of two vectors will be equal to the cosine of the angle between the vectors, times the lengths of each of the vectors.

In short, the dot product of two vectors, also known as their scalar product, is a way of multiplying vectors, arriving at a scalar quantity (in other words having a magnitude but no direction).

If both a and b are unit vectors, their dot product simply gives the cosine of the angle between them.

However, if 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(hat) as the unit vector in the direction of b is b / |b|.

[tram]

In short, it basically is a accepted way to get certain pieces of information about two geometric vectors.

You are not 'doing' anything to the vectors per se, merely carrying out a process to convert vector quantities to scalar quantities that can be measured and compared. If we did not have the dot product, we would not be able to compare one vector to another in any way.

[luffy]

Geometrically, the dot product (aka scalar product) of two vectors is the magnitude of one times the projection of the other along the first

Let's say we have two vectors, A and B.  The values |A| and |B| represent the lengths of vectors A and B, respectively, and Θ is the angle between the two vectors. The dot product of two vectors will be equal to the cosine of the angle between the vectors, times the lengths of each of the vectors.

In short, the dot product of two vectors, also known as their scalar product, is a way of multiplying vectors, arriving at a scalar quantity (in other words having a magnitude but no direction).

If both a and b are unit vectors, their dot product simply gives the cosine of the angle between them.

However, if 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(hat) as the unit vector in the direction of b is b / |b|.

Thanks a lot for your answer. I understand it a lot better now. Clearly, I am a noob on this forum, but I would give you karma if I could. Haha.

[samiira]

Geometrically, the dot product (aka scalar product) of two vectors is the magnitude of one times the projection of the other along the first

Let's say we have two vectors, A and B.  The values |A| and |B| represent the lengths of vectors A and B, respectively, and Θ is the angle between the two vectors. The dot product of two vectors will be equal to the cosine of the angle between the vectors, times the lengths of each of the vectors.

In short, the dot product of two vectors, also known as their scalar product, is a way of multiplying vectors, arriving at a scalar quantity (in other words having a magnitude but no direction).

If both a and b are unit vectors, their dot product simply gives the cosine of the angle between them.

However, if 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(hat) as the unit vector in the direction of b is b / |b|.

Thanks a lot for your answer. I understand it a lot better now. Clearly, I am a noob on this forum, but I would give you karma if I could. Haha.

nw

[dptjandra]

hehe...i'll do it

+1

[luffy]

hehe...i'll do it

+1

This is completely irrelevant, but I just read your signature. What ATAR score did you get? 99.95? lol...

[dptjandra]

hehe...i'll do it

+1

This is completely irrelevant, but I just read your signature. What ATAR score did you get? 99.95? lol...

yeh - i was very surprised when i got it actually...pleasantly surprised though

[luffy]

Yeah. Everyone is always telling me my ATAR will be based on my English Language study score and not my other subjects xD. I could do wonders if I got a 50 for that - Never gonna happen though. Haha.

[QuantumJG]

Hey guys, I have a very simple theory question,

What is the Dot Product? I don't mean "how do u find the dot product?" I mean, when you are finding the dot product of lets say vector 'a' and vector 'b'. What are you actually doing?

Many people have told me it is the "magnitude of a in the direction of b". However, I don't see how this makes sense, because if this was the case, then a.b = a.b(hat), which obviously is not true.

I just wanted to know, so that I will have a better understanding of 'applying' vectors.

So the dot product is very weird concept and in uni you will realize it is a special case of an inner product. The dot product can tell us if two vectors are orthogonal or parallel and to project one vector onto another. There are many other uses but these three are the main ones you will look at.

Let's look at some examples:

We can define the length of vector (usually denoted as a or | | ) :

a =

We can define an angle between two vectors, θ, as:

= ab cos(θ)

We can define a unit vector in the direction of vector as:



We can project onto by:

Proj

Note:

Proj - Proj = 0

[tram]

Hey guys, I have a very simple theory question,

What is the Dot Product? I don't mean "how do u find the dot product?" I mean, when you are finding the dot product of lets say vector 'a' and vector 'b'. What are you actually doing?

Many people have told me it is the "magnitude of a in the direction of b". However, I don't see how this makes sense, because if this was the case, then a.b = a.b(hat), which obviously is not true.

I just wanted to know, so that I will have a better understanding of 'applying' vectors.

So the dot product is very weird concept and in uni you will realize it is a special case of an inner product. The dot product can tell us if two vectors are orthogonal or parallel and to project one vector onto another. There are many other uses but these three are the main ones you will look at.

Let's look at some examples:

We can define the length of vector (usually denoted as a or | | ) :

a =

We can define an angle between two vectors, θ, as:

= ab cos(θ)

We can define a unit vector in the direction of vector as:



We can project onto by:

Proj

Note:

Proj - Proj = 0

haha i was contemplating whether or not i should explain the dot product in terms of it as a specific form of an inner product as it certainly helps explains what the dot product is.

[luffy]

Hey guys, I have a very simple theory question,

What is the Dot Product? I don't mean "how do u find the dot product?" I mean, when you are finding the dot product of lets say vector 'a' and vector 'b'. What are you actually doing?

Many people have told me it is the "magnitude of a in the direction of b". However, I don't see how this makes sense, because if this was the case, then a.b = a.b(hat), which obviously is not true.

I just wanted to know, so that I will have a better understanding of 'applying' vectors.

So the dot product is very weird concept and in uni you will realize it is a special case of an inner product. The dot product can tell us if two vectors are orthogonal or parallel and to project one vector onto another. There are many other uses but these three are the main ones you will look at.

Let's look at some examples:

We can define the length of vector (usually denoted as a or | | ) :

a =

We can define an angle between two vectors, θ, as:

= ab cos(θ)

We can define a unit vector in the direction of vector as:



We can project onto by:

Proj

Note:

Proj - Proj = 0

Thanks a lot for that.

However, understanding the alegbra is not too difficult. I was never really taught whats happening "geometrically" when u find the dot product and thats what I wanted to know.

But yea, you mentioned a lot of algebra to do with vectors, which I found both useful and interesting. Thanks.

[TrueTears]

Yeah agree with what QJP said, you will see that the dot product (or the Euclidean inner product) is just a special case of the inner product on R^n. In fact you can check that the Euclidean inner product satisfies the 4 axioms of an inner product on a real vector space V yourself.

[tram]

Yeah agree with what QJP said, you will see that the dot product (or the Euclidean inner product) is just a special case of the inner product on R^n. In fact you can check that the Euclidean inner product satisfies the 4 axioms of an inner product on a real vector space V yourself.

haha this is spech tt not uni maths.... not eveyone has the maths ability of you

[luffy]

Yeah agree with what QJP said, you will see that the dot product (or the Euclidean inner product) is just a special case of the inner product on R^n. In fact you can check that the Euclidean inner product satisfies the 4 axioms of an inner product on a real vector space V yourself.

Looks like there is a lot more to the dot product than I have come to realise. Clearly, my curiosity was justified. xD