Question

Let A , B and C be non-coplanar unit vectors , equally inclined to one another at an angle 'theta' .If:
A x B + B x C =pA +qB +rC , Find scalars p , q and r in terms of 'theta' .

Answer 1

I don't think I can solve this, but I am commenting so I get tagged on this question.

Answer 2

:)

Answer 3

we're given :
\[A\cdot A = B\cdot B = C\cdot C = 1\]
and
\[A\cdot B = B\cdot C = C\cdot A= \cos \theta \]

Answer 4

Next suppose the volume of parallelopiped formed by these 3 vectors as edges is \(V\), then we have :
\[A\cdot B\times C = B\cdot C\times A = C\cdot A\times B = V\]

Answer 5

using the given equation form 3 equations based on Volume and solve the system

you will get answer interms of V and keep in mind that the volume can be expressed in terms of \(\theta \)

Answer 6

@ganeshie8
The expression is not symmetric,
\(A\times B+B\times C \equiv(A-C)\times B = pA+qB+rC\)
which makes q=0.
I've come up with long expressions of each of A, B and C in terms of \(\cos(\theta)\), but was hoping to see some solutions in terms of, perhaps, vector spaces.

Answer 7

Ahh rest of it should be easy by interpreting the scalar triple product as volume of parallelopiped

The volume equals 0 if any two vectors are equal in the triple product

Answer 8

\[A \times B + B \times C =pA+qB +rC\]
we form 3 equations by dotting this both sides by A,B,C one by one :

\[A\cdot [A \times B + B \times C] =A\cdot [pA+qB +rC]\implies V = p +\cos t(q+r)\]
\[B\cdot [A \times B + B \times C] =B\cdot [pA+qB +rC]\implies 0 = q + \cos t(r+p)\]
\[C\cdot [A \times B + B \times C] =C\cdot [pA+qB +rC]\implies V = r + \cos t(p+q)\]

3 equations and 3 unknowns - we can solve them :)

Answer 9

@ganeshie8
It does look easy! Thanks.