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

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In summary, the volume of a parallelepiped with edges A, B, and C can be found by taking the cross product of B and C, which gives the area of a parallelogram, and then multiplying it by A, which represents the height. This can be shown analytically by using the dot product of A and the cross product of B and C. A helpful diagram is provided for visualization.
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tronter
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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].

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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?
 
  • #3
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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FAQ: How is the volume of a parallelepiped with edges A, B, and C calculated?

What is the formula for finding the volume of a parallelepiped?

The formula for finding the volume of a parallelepiped is V = l x w x h, where l is the length, w is the width, and h is the height.

How do you determine the length, width, and height of a parallelepiped?

The length, width, and height of a parallelepiped can be determined by measuring the three edges of the shape that are perpendicular to each other.

Can the volume of a parallelepiped be negative?

No, the volume of a parallelepiped cannot be negative. Volume is a measure of space and cannot have a negative value.

Can the volume of a parallelepiped be zero?

Yes, the volume of a parallelepiped can be zero if any of its dimensions (length, width, or height) is also zero. This would result in a flat, two-dimensional shape instead of a three-dimensional shape.

How is the volume of a parallelepiped different from a rectangular prism?

A parallelepiped is a three-dimensional shape with six parallelogram faces, while a rectangular prism is a shape with six rectangular faces. The formula for finding their volumes may be the same (V = l x w x h), but the shapes themselves are different.

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