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Find the volume of a parallelopiped havi...

Find the volume of a parallelopiped having three coterminus vectors of equal magnitude `|vec a|` and equal inclination `theta` with each other.

Text Solution

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The correct Answer is:
`|veca|^(3)sqrt(1+ 2cos theta ) (1 - cos theta)`
Let `veca, vecb and vecc` be three vectors of magnitude `|veca|` and equal inclination `theta` with each other.
volume of parallelepiped = `( veca. (vecb xx vecc) = [veca vecb vecc]`
`and [veca vecb vecc]^(2)= |{:(veca.veca,veca.vecb,veca.vecc),(vecb.veca,vecb.vecb,vecb.vecc),(veca.vecc, vecb.vecc, vecc.vecc):}|`
`|veca|^(2)|{:( 1, costheta, costheta),(costheta,1,costheta),(costheta, costheta, 1) :}|`
` |veca|^(6) ( 2cos^(3) theta- 3cos^(2)theta + 1) `
`|veca|^(6) (1-costheta)^(2) ( 1+ 2costhetaa)`
` or [veca vecb vecc] = |veca|^(3) sqrt(1 + 2cos theta) ( 1- cos theta)`
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