Home
Class 12
MATHS
Find the value of a so that the volume o...

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 ka n da hat i+ hat k` becomes minimum.

A

`-(1)/(sqrt(3))`

B

`(1)/(sqrt(3))`

C

`sqrt(3)`

D

`-sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
Given vecotors are `hati + lambda hatj + hatk, hatj+ lambda hat k and lambda hat i + hat k` which forms a parallelopiped
`:.` Volume of the parallelopiped is `V||{:(1,lambda,1),(0,1,lambda),(lambda,0,1):}||=1+lambda^(3)-lambda`
`rArr V= lambda^(3)-lambda+1`
On differentating. w.r.t. lambda, we get
`(dV)/(dlambda)=3lambda^(2)-1`
For maxima of minima, `(dV)/(dlambda)=0`
`rArr lamda+-(1)/(sqrt(3))`
`and (d^(2)V)/(dlambda^(2))=6lambda={{:(2sqrt(3)ge0,"for", lambda=(1)/(sqrt(3))),(2sqrt(3)le0, "for", lambda=-(1)/(sqrt(3))):}`
`:. (d^(2)V)/(d lambda^(2))` is poisitive for `lambda=(1)/(sqrt(3))`, so volume 'V' is minimum for `lambda=(1)/(sqrt(3))`
Promotional Banner

Similar Questions

Explore conceptually related problems

The volume (in cubic units) of the parallelopiped whose edges are represented by the vectors hat i + hat j , hat j + hat k and hat k + hat i is

Find the angle between the vectors hat i - 2 hat j + 3 hat k and 3 hat i - 2 hat j + hat k .

Prove that the vectors 3 hat i + hat j + 3 hat k and hat i - hat k are perpendicular.

Find the area of the parallelogram determined by the vectors vec a = hat i + hat j + 3 hat k , vec b = 3 hat i - 4 hat j + 5 hat k .

Find the volume of a parallelopiped with coterminous edges represented by the vectors 3 hat i + 4 hat j , 2 hat i + 3 hat j + 4 hat k and 5 hat k .

Find the volume of the parallelepiped whose coterminus edges are given by the vectors hat i + 2 hat j + 3 hat k , 2 hat i - hat j - hat k and 3 hat i + 4 hat j + 2 hat k

Find the value of lambda if the volume of a tetrashedron whose vertices are with position vectors hat i-6 hat j+10 hat k ,- hat i-3 hat j+3 hat k ,5 hat i- hat j+lambda hat ka n d7 hat i-4 hat j+7 hat k is 11 cubic unit.

Find the angle between the vectors hat i-2 hat j+3 hat ka n d3 hat i-2 hat j+ hat kdot

Find the area of the triangle whose adjacent sides are made by the vectors vec a = 3 hat i + 4 hat j + 4 hat k and vec b = hat i - hat j + hat k .

Find angle theta between the vectors vec a = hat i + hat j - hat k and vec b = hat i - hat j + hat k .