Vectors `vecA and vecB` satisfying the vector equation `vecA+ vecB = veca, vecA xx vecB =vecb and vecA.veca=1`. where `veca` and `vecb` are given vectors, are

Options

A. 1. `vecA = ((vecaxxvecb)-veca)/(a^(2))`
B. 2. `vecB = ((vecbxx veca) + veca (a^(2) - 1))/a^(2)`
C. 3. `vecA = ((vecaxxvecb)+veca)/(a^(2))`
D. 4. `vecB = ((vecbxx veca) - veca (a^(2) - 1))/a^(2)`

Answer

b,c,

Explanation

we have `vecA.vecB =veca`
`or vecA .veca + vecB.veca = veca.veca`
`or 1+ vecB.veca = a^(2)`
`or vecB.veca= a^(2)-1`
Also `vecA xx vecB =vecb`
`or veca xx (vecA xx vecB ) = veca xx vecb`
` or (veca. vecB)vecA - (veca.vecA) vecB = veca xx vecb`
`or (a^(2)-1) vecA -vecB = veca xx vecb`
( using (i) and ` veca. vecA =1`) (ii)
` and vecA + vecB =a`
form (ii) and (iii) , we have
`vecA= ((vecaxxvecb)+veca)/a^(2)`
`vecB=veca-{((vecaxxvecb)+veca)/a^(2)}`
` = ((vecbxxveca) + veca (a^(2) -1))/a^(2)`
thus `vecA= ((vecaxxvecb)+veca)/a^(2)`
` and vecB= ((vecbxxveca)+veca (a^(2) -1))/a^(2)`