Find a unit vector perpendicular to each of the vectors `(veca+vecb) and (veca-vecb)" where "veca=3hati+2hatj+2hatk and vecb=hati+2hatj-2hatk`.

Explanation

Given that `a=3hat(i)+2hat(j)+2hat(k) and b=hat(i)+2hat(j)-2hat(k)`.
`therefore a+b=(3hat(i)+2hat(j)+2hat(k))+(hat(i)+2hat(j)-2hat(k))=4hat(i)+4hat(j)+0hat(k)`
and `a-b=(3hat(i)+2hat(j)+2hat(k))-(hat(i)+2hat(j)-2hat(k))=2hat(i)+0hat(j)+4hat(k)`
Now , `(vec(a)+vec(b))xx(vec(a)-vec(b))=|(hat(i),hat(j),hat(k)),(4,4,0),(2,0,4)| = hat(i)(16-0)-hat(j)(16-0)+hat(k)(0-8)=16hat(i)-16hat(j)-8hat(k)`
`rArr |(vec(a)+vec(b))xx(vec(a)-vec(b))|=sqrt((16)^2+(-16)^2+(-8)^2)`
`=sqrt(256+256+64)=sqrt(576)=24`
`therefore` A unit vector , perpendicular to both `(a+b) and (a-b)` is `=((a+b)xx(a-b))/(|(a+b)xx(a-b)|)=(16hat(i)-16hat(j)-8hat(k))/(24)`
`=(8(2hat(i)-2hat(j)-1hat(k))/(24)=(1)/(3)(2hat(i)-2hat(j)-hat(k))`
`theefore` Required vector is either `(2)/(3)hat(i)-(2)/(3)hat(j)-(1)/(3)hat(k)`.