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B What is the real life utility of the dot product and cross product

  1. May 13, 2017 #1
    Today, my teacher asked us what is the real life utility of the dot product and cross product of vectors. Many of us said that one gives a scalar product, and one gives a vector product. But he said that, that was not the real life utility of the dot and cross product. He asked us, "Students, why do we have to learn these two concepts? Because they are given in the book? Or because they are important concepts of vectors, or refer do they have some actual utility."

    None of us could answer. Does anyone here know a proper answer to an improper question like this?

    N.B.: Please pardon me if it's a foolish question, and don't give me credit if it opens up a huge set of answers.
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  3. May 13, 2017 #2


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  4. May 13, 2017 #3
  5. May 13, 2017 #4
  6. May 13, 2017 #5


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    Is your class a physics class? If so, have you not yet studied concepts such as work (which is defined in terms of a dot product) and torque (which is defined in terms of a cross product)?
  7. May 13, 2017 #6
    Such answers were already given by us, but our teacher didn't accept them. We could not understand what he actually wanted.
  8. May 13, 2017 #7
    Technically, you could say that when you compute power of alternating current by voltage * current, you're doing dot product.
    As for cross product, various right-hand and left-hand rules of electromagnetism come to mind but that's hardly a real-life use.

    My bet is the teacher wanted to prevent you from asking that question. Best defense is offense, as they say.
  9. May 13, 2017 #8
    Sadly, I myself gave this answer, but was not accepted.

    I believe the teacher is trying to waste time by discussing such questions.
  10. May 13, 2017 #9


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    You might not realize that strategic sectors of engineering such as mechanical engineering and civil engineering are highly reliant on the application of dot products to calculate the component of forces exerted on various parts of mechanical and construction designs under actual operation. Without these, the lifetime and stability of mechanical products and buildings are hard to estimate.
  11. May 13, 2017 #10


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    I think you need to ask him what he means by "real life utility".

    Physics is "real life" to me. :oldwink:
  12. May 14, 2017 #11


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    As was mentioned earlier the dot product can give you the cosine of angle between two distinct non-zero vectors. It can also give you the projection of one vector on the other aka length of one vector times the cosine of the angle between them. Knowing the cosine you can determine if the non-zero vectors are parallel, anti-parallel, or perpendicular.

    The cross product can do similar feats or prestidigitation, namely it can give you the sine of the angle between two distinct non-zero vectors and whether they are parallel or perpendicular. It can also give you the area of a parallelogram whose two sides coincide with the vectors and it can give you a vector that is perpendicular to both non-zero vectors.

    It all depends on how you look at it and what you need it for. When I first encountered them is was somewhat mystified by the cross product understanding how to use it and its properties but understanding where it came from.

    Yet another insight is that they are somewhat complementary operations. The dot product gives you the projection of one vector on another whereas the cross product gives you that part which isn't the projection. In other words the two operations can be used to break up a vector into two components one along the other vector and one perpendicular to it.
    Last edited: May 14, 2017
  13. May 14, 2017 #12
    Same for me.

    After the class, what I understood: he himself could not find any proper meaning to "real life utility", and changed the question to "what are the applications of the two products", and this time there was none who couldn't answer.

    Sorry to waste the time of all of you behind such foolishness.
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