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Derive the expression for the volume of the prallelopiped whose coterminus edges are vectors bar(a),bar(b),bar(c). |
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Answer» SOLUTION :LET `bar(OA),barOBandbar(OC)` represent the coterminus edges `bar(a),bar(b)andbar(c)` respectively of the parallelopiped. Draw seg AN perpendicular to the PLANE of `bar(b)and bar(c)`. Let `theta` be the angle between `bar(b)andbar(c)andphi` be the angle between the line line AN and `bar(a)`. If `hat(n)` is the unit vector perpendicular to the palne of `bar(b)andbar(c)`, then the angle between `bar(a)andbar(c)`, then the angle between `bar(a)andhat(n)` is also `phi`. Volume of the parallelopiped = (area of parallelogram OBA'C) `xx` AN `=(bcsintheta)(acosphi)` `=a(bcsintheta)(cosphi)`. . . (1) Now let us consider the scalar TRIPLE produst `bar(a)*(bar(b)xxbar(c))` `bar(b)xxbar(c)=(bcsintheta)*hat(n):.|bar(b)xxbar(c)|=bcsintheta` `:.bar(a)*(bar(b)xxbar(c))=|bar(a)|*|bar(b)xxbar(c)|cosphi=a(bcsintheta)cosphi`. . . (2) `:.` from (1) and (2), volume of the parallelopiped `=bar(a)*(bar(b)xxbar(c))`
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