# Find the volume of the parallelepiped with one vertex at A(-1,-4,6)and adjacent vertices at B(-1,-5,4) C( 5,-11,-2) D( 3, -2, 9)

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### 1 Answer

You need to evaluate the vectors `bar(AB), bar(AC), bar(AD)` , such that:

`bar(AB) = (x_B - x_A)bar i + (y_B - y_A)bar j + (z_B - z_A) bar k`

`bar(AB) = (-1 + 1)bar i + (-5 + 4)bar j + (4 - 6) bar k`

`bar(AB) = - bar j - 2 bar k`

`bar(AC) = (x_C - x_A)bar i + (y_C - y_A)bar j + (z_C - z_A) bar k`

`bar(AC) = (5 + 1)bar i + (-11 + 4)bar j + (-2 - 6) bar k`

`bar(AC) = 6bar i -7 bar j - 8bar k`

`bar(AD) = (x_D - x_A)bar i + (y_D - y_A)bar j + (z_D - z_A) bar k`

`bar(AD) = (3 + 1)bar i + (-2 + 4)bar j + (9 - 6) bar k`

`bar(AD) = 4bar i + 2 bar j + 3bar k`

You need to evaluate the volume of parallelipiped, such that:

`V = bar(AD)*(bar(AC)Xbar(AB))`

You need to evaluate the cross product `(bar(AC)Xbar(AB))` such that:

`(bar(AC)Xbar(AB)) = [(bar i, bar j, bar k),(6,-7,-8),(0,-1,-2)]`

` `

`(bar(AC)Xbar(AB)) = 14 bar i - 6 bar k - 8 bar i + 12 bar j`

`(bar(AC)Xbar(AB)) = 6 bar i + 12 bar j - 6 bar k`

You need to evaluate the dot product` bar(AD)*(bar(AC)Xbar(AB))` , such that:

`bar(AD)*(bar(AC)Xbar(AB)) = <4,2,3>*<6,12,-6>`

`bar(AD)*(bar(AC)Xbar(AB)) = 4*6 + 2*12 + 3*(-6)`

`bar(AD)*(bar(AC)Xbar(AB)) = 30`

**Hence, evaluating the volume of parallelipiped, under the given conditions, yields **`V = 30.`