Dot product geometric proof question?

In summary, the conversation is discussing the proof of the dot product in three dimensions. The speaker is having trouble understanding the proof and presents their own 2D proof before attempting a 3D proof. They mention using the cosine rule approach but do not prefer it. The conversation also touches on the importance of intuition and understanding in mathematical proofs.
  • #1
fred4321
4
0
Dot product proof question?
Hi,

I'm having trouble understanding the proof of the dot product in three dimensions (not using the cosine rule approach).

Here's what I have for the 2D proof:
u = u1 i + u2 j
v = v1 i + v2 j
u.v = u1v1 + u2v2
u.v = |u| |v| cos(θ)
=> u1v1 + u2v2 = |u| |v| cos(θ)
x = XOV - XOU
=> u1v1 + u2v2 = |u| |v| cos(XOV - XOU)
u1v1 + u2v2 = |u| |v| (cos(XOV)cos(XOU) + sin(XOV)sin(XOU))
u1v1 + u2v2 = |u| |v| ( u2/|v| * u1/|u| + v2/|v| * v1/|u|)
u1v1 + u2v2 = |u| |v| ( u1u2/(|v||u|) + v1v2/(|v||u|) )
u1v1 + u2v2 = u1v1 + u2v2
Q.E.D.

Now when I go to do it in 3D:
u = u1 i + u2 j + u3 z
v = v1 i + v2 j + u3 z
u.v = u1v1 + u2v2 + u3v3
u.v = |u| |v| cos(θ)
u1v1 + u2v2 = |u| |v| ( sqrt(u1^2+v1^2)/|u| * sqrt(u2^2+v2^2)/|v| + u3/|u| * v3/|v|)
u1v1 + u2v2 = sqrt(u1^2+v1^2) * sqrt(u2^2+v2^2) + u3v3

and it doesn't seem to work out. The sqrt(...) parts are because I tried to find the angle between the vector and the x-y plane. I think I found the angle between the two vectors in three dimension incorrectly. Also, I reckon that I should be able to use my 2D proof for the 3D proof.

P.S. I am aware of the cosine rule approache to proving it, but I don't really like that method. There should be a way to go straight from:
u1v1 + u2v2 + u3v3
to
|u| |v| cos(θ)
 
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  • #2
I think of it this way. Say you want to take the dot product of some vector v with i. It's clear what it's doing. It just gives you the first component.

So, you take the projection of v1 onto i, which is what you want to prove. Then, it's clear what happens when you take the dot product of v1 with a multiple of i. Again, you get what you want. Same for v dot j and v dot k.

Now, if you want to take v1 dot some vector v2, well, v2 is just a combination of i, j, and k. And the dot product is distributive.

So, what you need is that if you project a vector onto a sum of two (or three, in this case) other vectors, it's the same thing as projecting onto each of the vectors and then adding the result.

And what does the thing you are trying to prove say?

It says that the dot product is obtained by projecting v1 onto v2 and then multiplying the lengths of the projection and v2.

So, that's basically it.

Maybe that's a little hard to follow. I leave it as an exercise to think about it until it makes sense.

Intuition and understanding is much more valuable than pushing symbols around. Once you understand, then you will know how to push the symbols around to prove what you want.
 
  • #3
I feel like I 'get' the dot product; my issue is, I can't seem to show it mathematically.
 
  • #4
But I think I did show it mathematically, though, if you pursue the argument all the way. Just focus on one step at a time. First, prove it when u is some vector and v = i. Just focus on that. One step at a time.
 

What is the dot product geometric proof?

The dot product geometric proof is a mathematical proof that demonstrates the relationship between two vectors in a geometric context. It shows how the dot product of two vectors can be calculated using their magnitudes and the cosine of the angle between them.

What is the purpose of the dot product geometric proof?

The purpose of the dot product geometric proof is to provide a visual and intuitive understanding of the dot product and its properties. It is often used in physics and engineering to explain vector operations and their applications.

What are the key components of the dot product geometric proof?

The key components of the dot product geometric proof are the two vectors, their magnitudes, and the angle between them. It also involves the use of trigonometric functions, specifically the cosine function, to calculate the dot product.

Why is the dot product important in mathematics and science?

The dot product is important in mathematics and science because it is a fundamental operation that is used to calculate the magnitude of a vector, the angle between two vectors, and determine if two vectors are perpendicular or parallel. It is also used in many applications, such as physics and engineering, to solve problems involving force, work, and energy.

Can the dot product geometric proof be extended to higher dimensions?

Yes, the dot product geometric proof can be extended to higher dimensions. In fact, the proof can be generalized to any number of dimensions using the concept of inner product spaces. This allows for the calculation of dot products in multidimensional vector spaces, which is useful in fields such as computer graphics and machine learning.

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