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In first-year physics classes, you learn how to add vectors using the "Tip-to-Tail" or "Parallelogram" method. In my Calculus 3 class we learned you can find the area of a Parallelogram by using the cross-product from 3 of its vertices. (A = |PQ x QR|)

My question is two fold:

1.) Is this PQR method of constructing a parallelogram the same thing as tip-to-tail method of adding vectors? i.e. You can use |PQ x QR| to find the area of a Tip-To-Tail parallelogram right?

2.) When would someone be interested in the area of a parallelogram that results from the addition of two vectors?

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    $\begingroup$ Are there any reasons or use cases for why you might be interested in the area of a parallelogram that results from the addition of two vectors? This is equivalent to asking whether vector cross products are used in physics. $\endgroup$
    – Ghoster
    Commented Mar 14, 2023 at 6:03
  • $\begingroup$ @Ghoster, maybe a better question is when would you want to do this. I will update question for clarification $\endgroup$
    – RudyJD
    Commented Mar 14, 2023 at 6:06

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  1. The parallelogram that shows up in the "tip-to-tail" method is the same parallelogram you find the area of using the cross product. However, these are distinct operations. One concerns the sum of two vectors, and the other a product between two vectors. It just happens that the same parallelogram shows up when you draw a picture.

  2. The area of a parallelogram is the magnitude of the cross product of the vectors that lie along two adjacent sides. It really has nothing to do with the sum of the two vectors. So your second question is closer to "When is the cross product useful in physics?". Here is an incomplete list of when you might want to use the cross product.

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  • $\begingroup$ OK so the magnitude of the cross product IS the area of a parallelogram constructed from its vectors. That satisfies my first question. I get that these are separate operations, but is there an example of using the cross product that is motivated by this connection tho? $\endgroup$
    – RudyJD
    Commented Mar 15, 2023 at 4:51
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    $\begingroup$ @RudyJD I don't know of an example that emphasizes this connection. Instead, I like to think that the two operations tell you different things about the same parallelogram: vector addition tells you about the diagonal of the parallelogram, and the cross product tells you about the area of the parallelogram. $\endgroup$
    – Aiden
    Commented Mar 15, 2023 at 5:05

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