# Find the value of ‘a’ so that the volume of parallelopiped formed by hat"i" + "a"hat"j" + hat"k", hat"j" + "a"hat"k" and "a"hat"i" + hat"k" becomes minimum. - Mathematics and Statistics

Sum

Find the value of ‘a’ so that the volume of parallelopiped formed by hat"i" + "a"hat"j" + hat"k", hat"j" + "a"hat"k" and "a"hat"i" + hat"k" becomes minimum.

#### Solution

Let bar"p" = hat"i" + "a"hat"j" + hat"k", bar"q" = hat"j" + "a"hat"k", bar"r" = "a"hat"i" + hat"k"

Let V be the volume of the parallelopiped formed by bar"p",bar"q",bar"r".

Then V = [bar"p"bar"q"bar"r"]

= |(1,"a",1),(0,1,"a"),("a",0,1)|

= 1(1 - 0) - "a"(0 - "a"^2) + 1(0 - "a")

= 1 + "a"^3 - "a"

∴ "dV"/"da" = "d"/"da"(1 + "a"^3 - "a")

= 0 + 3"a"^2 - 1 = 3"a"^2 - 1

and ("d"^2"V")/("da"^2) = "d"/"da"(3"a"^2 - 1)

= 3 × 2a - 0 = 6a

For maximum and minimum V, "dV"/"da" = 0

∴ 3a2 - 1 = 0

∴ "a"^2 = 1/3

∴ a = +- 1/sqrt3

Now, (("d"^2"V")/"da"^2)_("at a" = 1/sqrt3) = 6(1/sqrt3) = 2sqrt3 > 0

∴ V is minimum when a = 1/sqrt3

Also, (("d"^2"V")/"da"^2)_("at a" = - 1/sqrt3) = 6(- 1/sqrt3) = - 2sqrt3 < 0

∴ V is minimum when a = - 1/sqrt3

Hence, a = 1/sqrt3

Concept: Representation of Vector
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