Question

If A, B, C, D are (1, 1, 1), (2, 1, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Answer 1

Let `bara, barb, barc, bard` be the position vectors of points A(1, 1, 1), B(2, 1, 3), C(3, 2, 2) and D(3, 3, 4)

`bara = hati + hatj + hatk`

`barb = 2hati + hatj + 3hatk`

`barc = 3hati + 2hatj + 2hatk`

`bard = 3hati + 3hatj + 4hatk`

Given that vectors `bar(AB), bar(AC) and bar(AD)` represent the concurrent edges of a palallelopiped ABCD.

`bar(AB) = barb - bara = 2hati + hatj + 3hatk - hati - hatj - hatk = hati + 2hatk`

`bar(AC) = barc - bara = 3hati + 2hatj + 2hatk - hati - hatj - hatk = 2hati + hatj +hatk`

`bar(AD) = bard - bara = 3hati + 3hatj + 4hatk - hati - hatj - hatk = 2hati + 2hatj + 3hatk`

Consider, `bar(AB).(bar(AC)xxbar(AD))=|[1,0,2],[2,1,1],[2,2,3]|`

= 1(3 – 2) + 2(4 – 2)

= 1 + 4

= 5

Therefore, Volume of parallelopiped with AB, AC and AD as concurrent edges is 

V = `[bar(AB).(bar(AC) xx bar(AD))]` = 5 cubic units.