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Question

The value of 'a' so that the volume of parallel piped formed by $i with ˆ on top plus a j with ˆ on top plus k with ˆ on top comma j with ˆ on top plus a k with ˆ on top$ and $a with ˆ on top plus k with ˆ on top$ becomes minimum is

$- 3$
3
$\frac{1}{\sqrt{3}}$
$\sqrt{3}$

hint Hint:

We are given three sides of a parallelepiped. We have to find the value of "a" for which the volume is minimum. To find the minimum volume, we will differentiate the value of volume w.r.t the variable and equate it to zero.

The correct answer is: $fraction numerator 1 over denominator square root of 3 end fraction$

Let the given sides of parallelepiped be denoted as follows:
$A equals space i with hat on top space plus space a j with hat on top space plus space k with hat on top B equals space j with hat on top plus a k with hat on top C equals space a i with hat on top plus space k with hat on top$
The volume of a parallelepiped is given as $left square bracket space A with rightwards arrow on top space B with rightwards arrow on top space C with rightwards arrow on top right square bracket$ . It is scalar product of three vectors.
Now, substituting the values of all the vectors.
$V space equals open square brackets table row cell A with rightwards arrow on top end cell cell B with rightwards arrow on top end cell end table space C with rightwards arrow on top close square brackets space space space space space space equals space open vertical bar table row 1 a 1 row 0 1 a row a 0 1 end table close vertical bar space space space space equals space 1 left parenthesis 1 space minus 0 right parenthesis space minus a left parenthesis 0 space minus space a squared right parenthesis plus 1 left parenthesis 0 space minus space a right parenthesis space V space equals space 1 space plus space a cubed space minus space a space space space space space space space space space space space space space space space space space space space space space space$
Now, we will differentiate the equation of volume w.r.t the variable "a". To find any minimum value, we take the partial derivative and set it to zero.
$fraction numerator partial differential V over denominator partial differential a end fraction equals space 0 space plus space 3 a squared space minus space 1 W e space w i l l space s e t space t h e space v a l u e space o f space p a r t i a l space d e r i v a t i v e space t o space b e space z e r o space space space space 0 space equals space 3 a squared space minus space 1 3 a to the power of 2 space end exponent equals space 1 space space space a space space equals space square root of 1 third end root space space space a space equals space fraction numerator 1 over denominator square root of 3 end fraction$
This is the value of a for which the volume of parallelepiped is minimum.

For such questions, we should know the volume of parallelepiped.

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