Question

Given vectors a=1/2i +√3/2j
And b=√3/2j+1/2k
And c=i+2j+3k .The volume of parallelepiped with three coterminous edges as
u=(a•a)a +(b•b)b +(c•c)c
v=(a•b)a +(b•b)b +(b•c)c
w=(a•c)a +(b•c)b +(c•c)c equals??

Answer 1

@kainui

Answer 2

Actually i want to solve it without opening up a•b and a•c things!!
And then solving for u,v and w individually and hence solve the three vectors and then find the volume of parallelepiped having these as its edges..
I want it to solve by another way out .
Can anybody help pls??

Answer 3

need help

Answer 4

bud

Answer 5

I assume you know the volume is given by the scalar triple product
A dot (B cross C)
which is also the "determinant" of A,B, and C made into a 3x3 matrix

Answer 6

Yep ! I know !

Answer 7

@ganeshie8

Answer 8

@hartnn

Answer 9

the only idea I have is to use det(AB)= det(A)*det(B)

the volume of a parallelepiped with sides u, v, w is the scalar triple product
u x v dot w or if we make a matrix M with u, v, and w being the rows,
volume = det(M)

as u,v,w are linear combinations of a,b,c, we can write M as
M= [ a b c] M2 where [a b c] is a matrix with a,b,c being its columns, and M2 holds the coefficients
thus |M| = | [ a b c] | * det(M2)

Answer 10

I don't see enough structure in M2 to simplify it much, but maybe there is ?
for this problem a*a= 1 and b*b=1 , so I would use that simplification

Answer 11

Yeah ! I have done like this bt when we solve for a•c, b•c and a•b v would get a very yuckkkk term and then it would be further multiplied with a,b and c so this is the worst part here!!

Answer 12

@phi

Answer 13

@vishweshshrimali5

Answer 14

yea

Answer 15

The volume of the parallelepiped determined by the vectors a, b, and c is \(|c \cdot (a \times b)|\)

Answer 16

I know that bt i want to simplify the expression that r coming out to b very weird indeed!
Can't we solve that in a very simplified manner..
When v would be just finding the dot product of a and b and b and c and a and c we would get terms to be very disgusting and not only this we would have to multiply them to a ,b and c once again to get u,v and w...

Answer 17

I'm not sure what you are trying to do. Are you trying to find a formula for any 3 vectors a,b,c that then form u,v, and w ?

btw, for this problem I'm getting \[ 13 \sqrt{3} -\frac{151}{8} \]
which is not a very pretty number.