Volume of parallelepiped(Linear Algebra)

In summary, to find the volume of a parallelepiped with adjacent sides u, v, and w, take the absolute value of the dot product of u and the cross product of v and w. The dot product of two vectors is equal to the area of a parallelogram, and the height of the parallelepiped can be found by projecting a vertical line to the line orthogonal to the two vectors.
  • #1
Sheneron
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Homework Statement



Show that the volume of a parallelepiped having u,v, and w as adjacent sides is given by abs(u*(vXw)).

(The star means dot product)

I am not really sure how to solve this problem. I was thinking that Volume would equal norm(u)+norm(v)+norm(w), but I couldn't figure out anywhere to go from there. Can someone help me start this problem. Thanks.
 
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  • #2
First you should realize the dot product of 2 vectors is also the area of a parallelogram. Then you should note that the height of the parallelepiped is the projection of the vertical line to the line orthagonal (this should give you a hint) of the 2 vectors.
 
  • #3
Awesome, thanks
 

Related to Volume of parallelepiped(Linear Algebra)

What is a parallelepiped?

A parallelepiped is a three-dimensional figure with six faces, each of which is a parallelogram. It is similar to a rectangular prism, but its faces are not necessarily rectangles.

How is the volume of a parallelepiped calculated?

The volume of a parallelepiped is calculated by multiplying the area of the base by the height. The base is the area of one of the parallelogram faces, and the height is the perpendicular distance between the base and its opposite face.

What is the formula for finding the area of a parallelogram?

The formula for finding the area of a parallelogram is base times height. This can also be written as length times width.

Does the shape of a parallelepiped affect its volume?

Yes, the shape of a parallelepiped can affect its volume. For example, a longer and thinner parallelepiped will have a smaller volume than a shorter and wider one, even if they have the same base area.

Can the volume of a parallelepiped be negative?

No, the volume of a parallelepiped cannot be negative. Volume is a measure of space, and it cannot be negative. If the calculations for finding the volume result in a negative number, then there is an error in the measurement or calculation.

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