Volume of Paralellepiped (Calc3)

[rtareen]

This is about solving for the volume of a parallelepiped spanned by 3 vectors.

The area of the base of a parellepiped spanned by vectors v, w, and u is the area of the parallelogram spanned by v and w so it is |v x w|. Then the height is |u|cosθ. Why is this the height? It is the projection of u onto v x w. Why wouldn’t the height be |u|, instead of |u|cosθ? It seems like we’re taking the volume of a weird cube thing and making it even weirder by projecting the height by an angle.

I understand that the volume of a parallelepiped is easily given by det(u,v,w), but this is only the case because it can be written as the dot product

u • (v x w) = |v x w| (|u|cosθ) = b(h)

my question is why does h = |u|cosθ

 

[Dr.Peterson]

The "height" in the formula for volume of a parallelepiped must be perpendicular to the "base". That's exactly what the projection of u on v x w is! (This is a special case of the general formula for the volume of a prism, V = Bh where B is the area of the base and h is the perpendicular height.)

The idea here is essentially the same as the fact that the area of a parallelogram is not just ab (the product of the sides) but bh (the product of the base and the height measured perpendicular to the base). In fact, there, too, you could describe h as a projected on a perpendicular to b.

 

[rtareen]

The "height" in the formula for volume of a parallelepiped must be perpendicular to the "base". That's exactly what the projection of u on v x w is! (This is a special case of the general formula for the volume of a prism, V = Bh where B is the area of the base and h is the perpendicular height.)

The idea here is essentially the same as the fact that the area of a parallelogram is not just ab (the product of the sides) but bh (the product of the base and the height measured perpendicular to the base). In fact, there, too, you could describe h as a projected on a perpendicular to b.

Thank you Dr. Peterson. This is the most clear cut answer I have gotten for this question!