Find the value of `x\ a n d\ y` if the vectors ` vec a=3 hat i+x hat j- hat k\ a n d vec b=2 hat i+ hat j+y hat k` are mutually perpendicular vectors of equal magnitude.

Explanation

Given that,
` vec a=3 hat i+x hat j- hat k\ a n d vec b=2 hat i+ hat j+y hat k`
both are mutually perpendicular vectors of equal magnitude.
so, `vec a*vec b=0`
we know that,
`vec a*vec b=| veca||vec b| cos pji `
`phi= pi/2`
`=>(3 hat i+x hat j- hat k)*(2 hat i+ hat j+y hat k)=0`
`=>6-x-y=0`
`=>x+y=6......(1)`
and also according to question,
` | vec a|=| vec b|`
`=>sqrt((3)^2+(x)^2+(-1)^2)=sqrt((2)^2+(1)^2+(y)^2)`
`=>sqrt(10+x^2)=sqrt(5+y^2)`
`=>1=+x^2=5+y^2`
`=>x^2-y^2=5-10`
`=>(x-y)(x+y)=-5`
`=>x-y=-5/(x+y)`
`=>x-y=-5/6.....(2)`
from `(1)` and `(2)` we get,
`2x=6-5/6`
`=(36-5)/6`
`=31/6`
`=>x=31/12`
`y=6-x`
`=6-31/12`
`=5/12`
therefore `x=31/12 , y=5/12`