Question

if A, B, C, D are (1, i, I), (2, l ,3), (3; 2, 2) and (3, 3, 4) respetivly., then find the volume of the parallepiped with AB, AC and AD as 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)

`bar a=hati+hatj+hatk`

`barb=2hati+hatj+3hatk`

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

`bard=3 hati+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)xxbar(AD))]=5 "cubic unit"`