Discussion Overview
The discussion revolves around the calculation of the area of a parallelogram formed by two vectors in a two-dimensional space, with some participants exploring the implications of using the cross product in three dimensions.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant introduces the formula for the area of a parallelogram using the cross product of vectors $\vec{PQ}$ and $\vec{PR}$, expressing uncertainty about the components of the vectors.
- Another participant questions whether the correct expression for the length of the cross product should include a third dimension, suggesting it should be $\sqrt{()^2 + ()^2 + ()^2}$.
- Several participants provide coordinates for points in a two-dimensional space and calculate the vectors $\vec{AB}$ and $\vec{BC}$, arriving at a numerical result for the area using a determinant approach.
- One participant critiques the use of "short cut" notation for the cross product, emphasizing that it is only defined in three dimensions and suggesting that the vectors should be represented in three-dimensional form to properly compute the cross product.
- A later reply affirms the previous points about dimensionality and notation, indicating a shared understanding of the need for three-dimensional representation.
Areas of Agreement / Disagreement
Participants express differing views on the appropriate dimensionality for calculating the area of the parallelogram, with some advocating for two-dimensional representations and others emphasizing the necessity of three-dimensional vectors for the cross product.
Contextual Notes
The discussion highlights limitations in the assumptions about dimensionality and notation, particularly regarding the use of the cross product in two versus three dimensions.
