Deriving the algebraic definition the dot product

In summary, the dot product of two vectors can be defined geometrically as the product of their lengths and the cosine of the angle between them. This can be derived without using the law of cosines by projecting one vector onto the other and multiplying their lengths. In Euclidean geometry, this is equivalent to using the Pythagorean Theorem. The dot product can also be calculated algebraically by taking the sum of the products of the corresponding components of the vectors. This definition only applies to vectors in a Euclidean space.
  • #1
autodidude
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Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?
 
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  • #2
What exactly do you mean by "the geometric definition"? The simplest "geometric" definition I know is that the dot product of vectors u and v is the product of the length of u and the length of the projection of v on u, but I suspect that you are thinking of "length of u times length of v times cosine of the angle between them". Since that has "cosine" already in it, if not using the "cosine law" per se, you certainly will need to use something that has a "cosine" in it.

From the definition "length of u times the length of the projection of v on u", I would start by setting up a coordinate system in which the positive x-axis lies along vector u. Then u has components <a, 0>. Taking <b, c> to be the components of v, the "projection of v on u" is just <b, 0> so the dot product is ab= ab+ c0, the usual component formula for the dot product in this case. Get the general formula by rotating axes.
 
  • #3
Thanks HallsOfIvy, I'll try that.
 
  • #4
Well, in the 2-D case, given vectors "A", "B", with [itex](x_{a},y_{a})[/itex], [itex](x_{b},y_{b})[/itex], lengths a and b, respectively you may think in terms of right-angled triangles, and set up:
[tex]x_{a}=a\cos(\theta_{a})[/tex]
[tex]y_{a}=a\sin(\theta_{a})[/tex]
[tex]x_{b}=b\cos(\theta_{b})[/tex]
[tex]x_{b}=b\sin(\theta_{b})[/tex]
whereby we get:
[tex]x_{a}x_{b}+y_{a}y_{b}=ab\cos(\theta_{a}-\theta_{b})[/tex]
 
  • #5
First define the dot product for A and B to be the product of their magnitudes and the cosine of the angle between them.
We can see geometrically that A.(B + C) = A.B + A.C (think about the component of B and C along A), and therefore (A + B).(C + D) = A.C + A.D + B.C + B.D.
Choosing perpendicular axis, every vector can be written in terms of components, so A = a_1*i + a_2*j and B = b_1*i + b_2*j.
Therefore A.B = a_1*b_1*i.i + a_2*b_2*j.j + a_1*b_2*i.j + a_2*b_1*j.i.
Because i and j are unit vectors and perpendicular, i.i = 1, j.j = 1, i.j = 0, j.i = 0.
So we are left with A.B = a_1*b_1 + a_2*b_2.
 
  • #6
Also one thing to be aware of is that the algebraic defintion for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k.

and of course here's more info on it from wikipedia:

http://en.wikipedia.org/wiki/Vector_dot_product
 
  • #7
jedishrfu said:
Also one thing to be aware of is that the algebraic defintion for vector dot and cross products only work when you have your vectors defined in a Euclidean space like our old favorite x,y,z or i,j,k.

and of course here's more info on it from wikipedia:

http://en.wikipedia.org/wiki/Vector_dot_product
Strictly speaking the term "dot product" is only used Euclidean space. In other vector spaces the term is "inner product". Of course, any n-dimensional vector space is isomorphic to Rn so the two work out to be "essentially" the same.

And the cross product is only defined for R3, not general Euclidean spaces.
 
  • #8
Its also interesting to note the beauty of the inner and outer products. The inner product is a projection of one vector on another and the outer product is the non-projectable component of one vector on another.

Since the projection interpretation is valid for either vector then the resultant vector must be perpendicular to both and thus it becomes normal to the plane containing the two vectors.
 
  • #9
Couldn't we just always work a coordinate system where the x-axis is parallel to the x-component of one vector?
Or is that what you mean when you said 'get the general formula by rotating axes?'. And would the formula only work in two dimensions? [strike]If so, what about higher dimensions? [/strike]

EDIT: Woops, didn't see the new posts! SO for rotating the axes, would it much more complicated in 3D?

Thanks arildno, when you times the multiply the components and add them up, that just comes from the definition of dot product that HallsOfIvy mentioned right? ('length of u times the length of the projection of v on u').
 
  • #10
autodidude said:
Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?

Project the vector A onto the line through B. Multiply the length of this projection by the length of B. By similar triangles this is the same as projecting the vector B onto the line through A and multiplying the length of the projection by the length of A.

Call these products the dot product of A and B.

Either product divided by AB gives you the same number which depends only on the lines and not on the particular vectors.

Call this number the cosine of the angle between the vectors.

P.S. In Euclidean geometry the law of similar triangles which is what is used here is logically equivalent to the Pythagorean Theorem so using the Law of Cosines is really no different than what is posted here.
 

What is the algebraic definition of the dot product?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors as input and produces a scalar value as output. The algebraic definition of the dot product is the sum of the products of the corresponding components of the two vectors. In other words, it is the multiplication of the first component of the first vector with the first component of the second vector, added to the multiplication of the second component of the first vector with the second component of the second vector, and so on.

How is the dot product calculated?

To calculate the dot product of two vectors, you first multiply the corresponding components of the two vectors, and then add all of those products together. For example, if you have two vectors, A = (a1, a2, a3) and B = (b1, b2, b3), the dot product can be calculated as a1b1 + a2b2 + a3b3.

What is the purpose of the dot product?

The dot product has several important applications in mathematics and physics. It can be used to calculate the angle between two vectors, determine if two vectors are perpendicular, and project a vector onto another vector. It also has applications in fields such as computer graphics, engineering, and economics.

How is the dot product related to the magnitude and direction of vectors?

The dot product is related to the magnitude and direction of vectors in several ways. For example, if the dot product of two vectors is zero, it means that the vectors are perpendicular to each other. If the dot product is positive, it means that the vectors are pointing in the same general direction, and if it is negative, they are pointing in opposite directions. The magnitude of the dot product also gives information about the length of the projection of one vector onto another.

What are some properties of the dot product?

The dot product has several properties, including commutativity (A · B = B · A), distributivity (A · (B + C) = A · B + A · C), and associativity with scalar multiplication (k(A · B) = (kA) · B = A · (kB)). It is also related to the magnitude of vectors, as the dot product of a vector with itself is equal to the square of its magnitude. Additionally, the dot product is used to define the concept of orthogonality, where two vectors with a dot product of zero are considered orthogonal or perpendicular to each other.

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