How is the volume of a parallelepiped with edges A, B, and C calculated?

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Homework Help Overview

The discussion revolves around calculating the volume of a parallelepiped defined by three edges, A, B, and C. The original poster presents a statement regarding the volume formula involving the cross product of vectors B and C.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the relationship between the cross product and the area of a parallelogram, questioning how this relates to the volume calculation. There is also a discussion about the role of vector A as the height in the context of the volume formula.

Discussion Status

Some participants have offered insights into the geometric interpretation of the volume formula, including the relationship between the height and the angle between vectors. A diagram has been shared to aid visualization, indicating a productive exchange of ideas.

Contextual Notes

There is a mention of angles and geometric relationships that may require further clarification. The original poster's approach and assumptions about the vectors involved are also under consideration.

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Homework Statement



Show that the volume of a parallelepiped with edges [tex]A,B,C[/tex] is given by [tex]A \cdot (B \times C)[/tex].

Homework Equations


The Attempt at a Solution



[tex]B \times C[/tex] is the area of a parallelogram. From here I would I deduce the above result?
 
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Is [tex]A[/tex] just the height?
 
You're right, in two dimensions x and y the area is A=|bxc|. The height is actually |a|*|cos(theta)|. Where theta is the angle between vector a, and the cross product of vectors b and c. I'll include a picture but I hope this helps analytically to prove it. V=|a dot product to (b cross product with c.
 

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