Question

Prove that volume of a parallelopiped with coterminal edges as `bara,barb,barc` is `[bara,barb,barc]`. Hence find the volume of the parallelopiped with coterminal edges `hat("i")+hat(j),hat(j)+hat(k)andhat(k)+hat("i")`.

Answer 1

Let the coterminal edges of a parallelopiped are `vec(a),vecb,andvecc` respectively.

`vecn=vecbxxvecc=|vecbxxvecc|hat(n)`
= (Area of parallelogram OBDC) `hat(n)`
`=|vecb||vecc|sinthetahat(n)`
Then `vec(a)*(vecbxxvecc)=|veca||vecbxxvecc|cosphi`
`=|veca||vecb||vecc|sinthetacosphi`
`=|vecb||vecc|sinthetasintheta.|veca|cosphi`
= (Area of parallelogram OBDC)x (Height of side OA)
`[{:(veca,vecb,vecc):}]`= Volume of parallelopiped
Given, `veca=hat("i")+hat(J)`,
`vecb=hat(J)+hat(k)`
`vecc=hat(k)+hat("i")`
`V=[{:(veca,vecb,vecc):}]=|{:(1,1,0),(0,1,1),(1,0,1):}|`
`=1(1-0)-1(0-1)+0`
`=1+1`
= 2 unit