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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.
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Solution
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.
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