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")`.

Explanation

Let `vec(OA), vec(OB) and vec(OC)` represent the coterminus edges `bara, barb and barc` respectively of the parallelo-piped. Draw segment AN perpendicular to the plane of `barb and bar c`. Let `phi` be the angle between `barb and bar c` and `phi` be the angle between the line AN and `bara.`

`therefore(AN)/(OA) = cos phi rArr AN = OA cos phi = a cos phi" ...(i)"`
If `hatn` is the unit vector perpendicular to the plane of `barb and barc`, then the angle between `bara and hatn` is also `phi`.
`therefore" Volume of the parallelopiped "`
= Area of parallelogram `(OBA'C xx AN)`
`= bc sin theta xx a cos phi" [Using (i)]"`
`= a (bc sin theta) cos phi" ...(ii)"`
Now, let us consider the scalar triple product
`[bara barb barc]=bara .(barb xx barc)`
`barb xx barc=(b c sin theta)hatn`
`therefore" "|barb xx barc|=bc sin theta`
`therefore" "[bar a bar b barc]= bara . (barb xx barc)`
`|bara |.|barb xx barc| cos phi" ...(iii)"`
From equations (ii) and (iii).
Volume of the parallelopiped `=[bara barb barc]` ltBrgt Given `bara=hati+hatj, barb = hatj + hatk and barc = hatk + hati`
`therefore" "[bara bar b bar c]=[(1,1,0),(0,1,1),(1,0,1)]`
`=1(1-0)-1(0-1)+0(0-1)`
`=1+1+0=2`
`therefore" The volume of the parallelopiped = 2 cubic units."`