Find a unit vector perpendicular to the plane ABC, where the coordinates of `A ,\ B ,\ a n d\ C` are `A(3,\ -1,\ 2),\ B(1,\ -1,\ -3)a n d\ C(4,\ -3,\ 1)dot`

Explanation

Given coordinates of `A ,\ B ,\ a n d\ C`
are `A(3,\ -1,\ 2),\ B(1,\ -1,\ -3)a n d\ C(4,\ -3,\ 1)dot`
`vec (AB) = vec A - vec B `
`=(3,-1,2)-(1,-1,-3)`
`= (2,0,5)`
`vec(BC) = vec B - vec C`
`=(1,-1,-3)-(4,-3,1)`
`=(-3,2,-4)`
`vec (AB)xx vec (BC)= |(hat i , hat j, hat k),(2,0,5),(-3,2,-4)|`
`=hat i (0-10)-hat j (-8+15)+hat k (4-0)`
`=-10 hat i -7 hat j +4 hat k `
` |vec (AB)xx vec (BC)|=sqrt(100+49+16)`
`=sqrt(165)`
so, unit vector perpendicular to the plane ABC is ,
`(-10 hat i -7 hat j +4 hat k)/sqrt 165`