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Math Help - Vector Dot and Cross Products

  1. #1
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    Question Vector Dot and Cross Products

    OK I'm pretty new to Vectors and I'm missing something very basic here.

    Any help would be appreciated.
    I can recite equations on both Dot and Cross products until I am blue in the face.

    I can even understand where the Dot product comes from in terms of proving it using triangles and the Cosine Rule.

    The question I'm struggling to answer is exactly what are these products showing me and why in layman's terms?

    I've spent 2 days solidly reading everything I can on this and haven't found a satisfactory answer.

    My best guesses are as follows:-
    1) For the Dot product, you are essentially applying two forces and working out a combined effect of both of them. Clearly the maximum effect is if both of these forces are in the same direction. As you move one of them away you can use triangles to work out the effective "strength" of the second force which will be between 0 and 1 times it's force magnitude.
    This answer can't be right because at 90 degrees, the Dot Product is zero but if you had two forces at 90 degrees of the same magnitude clearly there would be movement at 45 degrees. What have I missed here?

    2) I suspect, the Cross Product is showing the resultant rotation from two forces acting on each other. Is this correct? If so, this may well help explain why we use sine theta because that would be the force line of force one exerted on force two with a rotation about the point where the vectors tails are. How does this work with displaced vectors though is they are NOT tail to tail?

    I'm sure I'm miles out with my thoughts but I don't want anyone thinking I haven't at least had a go at this myself.

    I particularly want to know this because my interest is in Electromagnetics and I need to userstand this before I start to move onto Div, Curl an d other frightening things
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  2. #2
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    [QUOTE=oaksoft;85214]
    Quote Originally Posted by oaksoft View Post
    OK I'm pretty new to Vectors and I'm missing something very basic here.

    Any help would be appreciated.
    I can recite equations on both Dot and Cross products until I am blue in the face.

    I can even understand where the Dot product comes from in terms of proving it using triangles and the Cosine Rule....
    Hello,

    I'll try to give you a very basic geometric explanation about the 2 vector products.

    1. The result of the dot product of two vectors is a real number.
    The result of the cross product of two vectors is a vector.

    2. Dot product:

    If you have 2 vectors \vec a, \vec b then the dot product is:

    \vec a \cdot \vec b = |\vec a  | \cdot |\vec b| \cdot \cos(\alpha)

    Since |\vec a  | is the length of \vec a and |\vec b  | is the length of \vec b the product

    |\vec a| \cdot \cos(\alpha) is the length of \vec a projected perpendicular on the \vec b.

    And since the product of 2 numbers can be represented as the area of an rectangle the shaded rectangle represents the value of the dot product.

    3. Cross product.

    two vectors which are not-collinear ( don't have the same direction) span a plane P. The cross product of \vec a and \vec b gives a new vector which is perpendicular to both vectors that means the new vector is the normal vector of the plane (a plane has only one normal vector!):

    \vec n = \vec a \times \vec b

    The length of \vec n is calculated by:
    |\vec n|=|\vec a| \cdot |\vec b| \cdot \sin(\alpha) which describes the area of a parallelogram with |\vec a| and \vec b as it's sides.
    Attached Thumbnails Attached Thumbnails Vector Dot and Cross Products-vec_products.jpg  
    Last edited by earboth; November 16th 2007 at 04:32 AM.
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  3. #3
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    Hi. Thanks for having a go at helping but this doesn't really answer my questions.

    For example, I'm still none the wiser as to WHY the cross product creates a new perpendicular vector.

    Another example, WHY does the dot product need you to consider the portion of one vector imposed on the other?

    The problem here is that dot and cross product definitions seem to be presented as something which falls out of equations.
    What I'm looking for however is different.

    I think practical non-maths examples of use may help.

    Anyone know of any?
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  4. #4
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    Quote Originally Posted by oaksoft View Post
    Hi. Thanks for having a go at helping but this doesn't really answer my questions.

    For example, I'm still none the wiser as to WHY the cross product creates a new perpendicular vector.

    Another example, WHY does the dot product need you to consider the portion of one vector imposed on the other?

    The problem here is that dot and cross product definitions seem to be presented as something which falls out of equations.
    What I'm looking for however is different.

    I think practical non-maths examples of use may help.

    Anyone know of any?
    I think the best reason for why the dot and cross products are what they are is because they are useful. There are probably an infinitude of ways to define a product between two vectors.

    In Physics the first basic use of the dot product comes in the form of the work equation: W = \vec{F} \cdot \vec{s}. Work in this sense is more or less the "effort" required to accomplish a task, and it makes sense (when you think about it for a while) that the work done should depend on the size of the angle between the applied force and displacement. The use of the cosine function is pretty much a convenience, since it has the properties that are desirable. Even the concept of no work being done by an applied force that is perpendicular to a given displacement makes a certain amount of sense: the force gets nothing done.

    As for the cross product, the first place a Physics student sees it is the torque equation, but I think the best argument for its use is actually for angular momentum. I won't go into details (unless you really want them) but again there are sensible features to using the sine of the angle between the vectors and (if you use some imagination) a reason for the perpendicular vector nature of the result.

    I will note that the dot product between two vectors is slightly more general than the cross product because there is a very easy way to generalize the dot product to higher dimensions, but as far as I know the cross product as defined can only be used in 3D. (I believe that the problem is not that it can't be generalized, but that it can't be uniquely generalized and that there is more than one useful generalization, but please don't quote me on that.)

    -Dan
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  5. #5
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    There is a Lagrange identity that says,
    |\bold{a} \times \bold{b}|^2 = |\bold{a}|^2|\bold{b}|^2 - |\bold{a}\cdot \bold{b}|^2
    Thus, (using the geometric meaning of dot product)
    |\bold{a} \times \bold{b}|^2 = |\bold{a}|^2|\bold{b}|^2 - |\bold{a}|^2|\bold{b}|^2\cos^2 \theta
    Thus,
     |\bold{a} \times \bold{b}|^2 = |\bold{a}|^2|\bold{b}|^2\sin^2\theta
    Take square roots,
     |\bold{a} \times \bold{b}| = |\bold{a}||\bold{b}||\sin \theta|
    This means: The norm of a cross product of two (non-zero) vectors is the area of the paralleogram form. (Because the angle of parallelogram is from by multipling its sides by the sine of its angle).

    Now it can be easily shown that \bold{a} \cdot (\bold{a}\times \bold{b}) = 0 \mbox{ and }\bold{b}\cdot (\bold{a}\times \bold{b}) = 0. This means: The cross product is a vector perpendicular to two (non-zero) vectors.

    Putting it together we have: The cross product of two (non-zero) vectors is a vector perpendicular to both vectors whose length is the size of the parallelogram area form.

    Of course the only problem is what way do we take the cross product, i.e. up or down. But that is basically what the cross product means in geometric terms.
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  6. #6
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    Quote Originally Posted by topsquark View Post
    I think the best reason for why the dot and cross products are what they are is because they are useful. There are probably an infinitude of ways to define a product between two vectors.

    In Physics the first basic use of the dot product comes in the form of the work equation: W = \vec{F} \cdot \vec{s}. Work in this sense is more or less the "effort" required to accomplish a task, and it makes sense (when you think about it for a while) that the work done should depend on the size of the angle between the applied force and displacement. The use of the cosine function is pretty much a convenience, since it has the properties that are desirable. Even the concept of no work being done by an applied force that is perpendicular to a given displacement makes a certain amount of sense: the force gets nothing done.

    As for the cross product, the first place a Physics student sees it is the torque equation, but I think the best argument for its use is actually for angular momentum. I won't go into details (unless you really want them) but again there are sensible features to using the sine of the angle between the vectors and (if you use some imagination) a reason for the perpendicular vector nature of the result.

    I will note that the dot product between two vectors is slightly more general than the cross product because there is a very easy way to generalize the dot product to higher dimensions, but as far as I know the cross product as defined can only be used in 3D. (I believe that the problem is not that it can't be generalized, but that it can't be uniquely generalized and that there is more than one useful generalization, but please don't quote me on that.)

    -Dan
    I appreciate your response.
    Interestingly, things like these products tend to come from one of two places:-

    1) They have been specially devised to solve a specific problem.
    2) They have literally dropped out of an equation as interesting insights.

    I'm not sure which is which and I haven't read anything to persuade me which is correct. I suppose I'll just have to keep hunting around in the meantime. Thanks.
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