Let ` vec a=alpha hat i+2 hat j-3 hat k , vec b=alpha hat i+2alpha hat j-2 hat k ,a n d vec c=2 hat i-alpha hat j+ hat kdot` Find thevalue of `6alpha,` such that `{( vec axx vec b)xx( vec bxx vec c)}xx( vec cxx vec a)=0.`

Answer

4

Explanation

`veca=alphahati+2hati-3hatk,vecb=hati+2alphahati-2hatk,`
`vecc= 2hati - alphahatj + hatk {(veca xx vecb) xx (vecb xx vecc)}xx (vecc xx veca)=vec0`
`or {[veca vecb vecc] vecb-[veca vecbvecb]vecc}xx (veccxxveca)=0`
`or [veca vecb vecc] vecb xx (vecc xx veca)=vec0`
`or [veca vecb vecc] ((veca.vecb)vecc- (vecb.vecc)veca)=vec0`
`[ veca vecb vecc] =0 ` ( `veca and vecc` are not collinear)
`Rightarrow |{:(alpha,2,-3),(1,2alpha,-2),(2,-alpha,1):}|`
`or alpha(2alpha - 2alpha) -2(1+4) -3 (-alpha - 4alpha)=0`
`or 10 -15 alpha=0`
`alpha 2//3`