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

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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) = 4bar 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) = 2bar 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),(4,7,-8),(0,1,-2)]`

`(bar(AC)Xbar(AB)) = -14 bar i + 4 bar k + 8 bar i + 8 bar j`

`(bar(AC)Xbar(AB)) = -6 bar i + 8 bar j + 4 bar k`

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

`bar(AD)*(bar(AC)Xbar(AB)) = <2,-2,3>*<-6,8,4>`

`bar(AD)*(bar(AC)Xbar(AB)) = |2*(-6) + (-2)*8 + 3*4|`

`bar(AD)*(bar(AC)Xbar(AB)) = |-16| = 16`

**Hence, evaluating the volume of parallelipiped, under the given conditions, yields `V = 16.` **