Prove that: `(vecaxxvecb)xx(veccxxvecd)+(vecaxxvecc)xx(vecd xx vecb)+(vecaxxvecd)x(vecbxxvecc)=2[vecb vecc vecd] veca`

Explanation

`(veca xxvecb)xx(veccxxvecd)+(veca xx vecc)xx (vecd xx vecb) + (veca xx vecdD) xx (vecb xx vecc)`
Here ` (veca xx vecb) xx (vecc xx vecd)`
`-(veccxxvecd.vecb)veca + (veccxxvecd.veca)vecb `
`= [veca vecc vecd]vecc-[vecb vecc vecd]veca`
`(veca xx vecc)xx (vecd xx vecb)`
`=-(vecc xx vecd.vecb)veca + (veccxxvecd.veca)vecb`
`=[veca vecd vecb]vecc-[vec cvecd vecb] veca`
`(veca xx vecd) xx (vecb xx vecc)`
`= (vecaxx vecd.vecc)vecb- (veca xx vecd.vecb)vecc`
` =-[veca vecc vecd] vecb -[veca vecd vecb] vecc`
(Note, Here we have tried to write the given expression is such a way that we can get terms involving `veca` and other simil,ar terms which can get cancelled)
Adding (i), (ii) and (iii), we get
Given vector `=- 2 [vecb vecc vecd] veca = k veca`
Hence, given vector is parallel to `veca`.