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

u = <<2, 5,-8>>, << 3,0,4>>, << 0,10,-7>>

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

### 1 Answer | Add Yours

The volume will be the absolute value of the determinant of the matrix:

`V=|[2,5,-8],[3,0,4],[0,10,-7]|`

`=2(-40)-5(-21)+(-8)(30)`

=-215

So the volume is 215 cu units.

You can also compute the determinant:

(0+0-240)-(0+80-105)

=-240+25=-215

We can compute the volume of a parallelpiped formed by the span of three vectors a,b, and c by `V=|(a"x"b)*c|`

Here `{(2,5,-8) "x" (3,0,4) }* (0,10,-7)`

`=(20,-32,-15)*(0,10,-7)`

=0-320+105

=-215 as before.

Taking the absolute value yields 215 cu units.

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