similar to how the absolute value of * ab cd * gives the area of parellelogram spanned by

** << a,b >> and <<c,d >>, **

the absolute value of determinant * abc def ghi *gives the volunme of a parallelepiped spanned by

**<< a,b,c>> , << d,e,f>> , and << g, h,i>> **

in three dimensions.

find the valume of the parallepiped sanned by

<< 5,0,0>>, << 0,6,0>>, << 0,0,8>>

using the determinant formula, verify answer by using more familiar formula

### 1 Answer | Add Yours

Find the volume of the parallelpiped formed by the span of <5,0,0>,<0,6,0>,<0,0,8>:

The volume is the determinant:

`|[5,0,0],[0,6,0],[0,0,8]|`

`=5|[6,0],[0,8]|` (the other minors have coefficient zero)

=5(48-0)

=240

** Another way to find the determinant of a 3x3 matrix:

=(5*6*8+0+0)-(0+0+0)

=240

=200

The volume of a parallelpiped is also `V=l*w*h` ; in this case we get V=5(6)(8)=240

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The volume is 240 cubic units

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