Let `vec(a)` and `vec(b)` be two unit vectors such that `vec(a).vec(b)=0` For some `x,y in R`, let `vec(c)=xvec(a)+yvec(b)+(vec(a)xxvec(b))` If `|vec(c)|=2` and the vector `vec(c)` is inclined at same angle `alpha` to both `vec(a)` and `vec(b)` then the value of `8cos^2alpha` is

Answer

3

Explanation

We have
`vec(c ) = x vec(a) +y vec(b) + vec(a) xx vec(b) " and " vec(a) "." vec(b) =0`
`|vec(a)|=|vec(b)|=1 " and " |vec(c ) |=2`
Also given `vec( c) ` is inclined on `vec(a) " and " vec(b)` with same angle `alpha`.
`:. ,vec(a)"." vec(c ) = x|vec(a)|^(2) +y(vec(a)"."vec(b))+ vec(a)"."(vec(a) xx vec(b))`
`|vec(a)||vec(c )| cos alpha =x +0+0`
Similarly
`|vec(b)||vec(c)| cos alpha=0 +y+0`
`rArr y=2 cos alpha`
`|vec(c )|^(2) =x^(2) +y^(2) +|vec(a)xx vec(b)|^(2)`
`4=8 cos^(2) alpha+ |alpha|^(2) |vec(b)|^(2) sin^(2) 90^(@)`
`4=8 cos^(2) alpha+1 rArr 8 cos^(2) alpha=3`