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प्रश्न
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.
उत्तर
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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