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Cross Product vs. Dot product

  1. May 23, 2012 #1
    Why is dot product given by a*b cosθ whereas cross product ab sinθ.
     
  2. jcsd
  3. May 23, 2012 #2
    Can you elaborate? Neither the dot product nor the cross product are "given" by either of those expressions.
     
  4. May 23, 2012 #3
    The cross product is a vector quantity so only the magnitude is given by a*b*sin θ. That does give you the dot product, but what exactly do you mean by "why"? Are you looking for a rationale for why mathematicians created such a thing? There are a lot of operations in math which were created because they are useful. And sometimes they are just created just purely as an intellectual curiosity.
     
  5. May 24, 2012 #4

    K^2

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    Dot product is a linear map. It's really a whole lot more complicated once you look at how dual vector spaces work, but lets forget that for a moment. Focus on linearity. You want property that a.b+a.c = a.(b+c) That tells you right away that a.(2b) = 2 a.b In other words, the result must be proportional to magnitude of b. Similarly, you can show that it's proportional to a. So you know the format must be ||a|| ||b|| * something. That something can only depend on directions of the vectors.

    Next, if you dig a bit deeper into linear algebra, you'll see that (Ua).(Ub) = a.b, where U is any unitary transform. That is, if for example, you rotate both a and b vectors, their dot product doesn't change. That tells you that the dot product can only depend on the angle between the two vectors, not individual orientations. So you know that a.b = ||a|| ||b|| f(θ).

    Showing that f(θ)=cos(θ) requires one more constraint. We want to make sure that length of our vectors is defined by the dot product itself. That is ||a||²=a.a That requires a function that satisfies f(0)=1, and then with some more hairy math you can get it to cos(θ). Unfortunately, I don't know a simple way to prove it that doesn't involve complex numbers.
     
  6. May 24, 2012 #5
    OK leave it. Please explain why do we use the formula,

    θ = cos^-1[vector a * vector b/ a * b]

    to find the angle between two vectors a and b.
     
    Last edited: May 24, 2012
  7. May 24, 2012 #6
    Physics Kiddy
    I see that you are still in high school. It's good that you are showing an interest is this so soon.
    We can add, subtract and multiply ordinary numbers, and each of these operations are unique. There is only one way to multiply two numbers.

    With vectors it is a little different. There is still only one way to add and subtract vectors (head to tail scheme you may be familiar with) but there are several different ways to multiply vectors.

    In both cases we are multiplying their magnitudes (this makes sense right).
    Here is the rationale for the difference:

    First dot product:
    If the vectors are pointing in the same direction the dot product gives you the full product of their magnitudes |A||B| since cos(0) =1. If the vectors are not pointing in the same direction the dot product gives you less. When they are pointing at 90 degree angles to each other it gives you zero. In other words the dot product is the actual product but reduced as the angle between the vectors increases. Imagine pulling a wagon by its handle. You are trying to pull it horizontally, but the handle is at some angle. The steeper the angle the less of your pulling effort works horizontally. If the angle is at 90 degrees (straight up) you will not be able to pull the wagon horizontally no matter how hard you pull. You will be able to pull it the easiest if the handle is horizontal (angle 0). This is an example of use of dot product.

    Cross product:
    Here if the vectors are at 90 degree angles to each other you get the full product of their magnitudes |A||B|. As the angle *decreases* you get less, and at 0 degrees you get 0 (just opposite of dot product). Imagine pulling on a big wrench in order to twist off a tight bolt. Your best bet is to pull in a direction that is at a right angle to the wrench. If you pull at a smaller angle, less of your effort will be torquing the bolt. If you pull at a 0 degree angle (along the length of the wrench) none of your effort is working to untwist bolt. This is an application where cross product is used.

    You can rearrange dot product formula from:
    A dot B = |A||B|cosθ
    to
    θ = cos^-1[(A dot B)/|A||B|]
    Which is convenient for finding angle given two vectors.
     
    Last edited: May 24, 2012
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