Consider the vectors ` hat i+cos(beta-alpha) hat j+cos(gamma-alpha) hat k ,cos(alpha-beta) hat i+ hat j+"cos"(gamma-beta) hat k` and `cos(alpha-gamma) hat i+cos(beta-gamma) hat k+a hat k `where `alpha,beta`, and `gamma` are different angles. If these vectors are coplanar, show that `a` is independent of `alpha,beta` and `gamma`

Explanation

Since the vectors are coplanar, we have
`" "|{:(1,,cos(beta-alpha),,cos(gamma-alpha)),(cos(alpha-beta),,1,,cos(gamma-beta)),(cos(alpha-gamma),,cos(beta-gamma),,a):}|`
`" "|{:(cosalpha,,sinalpha,,0),(cosbeta,,sinbeta,,0),(cosgamma,,singamma,,a-1):}||{:(cosalpha,,sinalpha,,0),(cosbeta,, sinbeta,,0),(cosgamma,,singamma,,1):}|=0`
`rArr a=1`