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Derive the expression for the volume of ...

Derive the expression for the volume of the prallelopiped whose coterminus edges are vectors `bar(a),bar(b),bar(c)`.

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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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