Find the value of `a` so that the volume of the parallelepiped formed by vectors ` hat i+a hat j+k , hat j+a hat k and a hat i+ hat k` becomes minimum.

Options

A. `-3`
B. 3
C. `1//sqrt3`
D. `sqrt3`

Answer

c

Explanation

Volume of parallelepiped fromed by ` vecu =hati + a hatj +hatk.`
` vecv = hatj +ahatk , vecw =a hati +hatk` is
`v=[vecu vecv vecw]= |{:(1,a,1),(0,1,a),(a,0,1):}|`
` 1 ( 1-0) -a (0-a^(2))+1 (0-a) `
` 1+ a^(3) -a`
For V to be minimum , `(dV) /(da) =0`
` RIghtarrow 3a^(2) -1 =0`
`or a = +- 1/sqrt3`
But ` a gt 0 or a = 1/sqrt3`