Find the unit vector in the direction of vector `overline(PQ)`, where P and Q are the points `P=(1,2,3).Q=(4,5,6)` respectively.

Explanation

The given points are `P(1,2,3) and Q(4,5,6)`.
`therefore x_(1)=1,y_(1)=2,z_(1)=3 and x_(2)=4,y_(2) =5, z_(2)=6`
So vector `PQ=(x_(2)-x_(1))hat(i)+(y_(2)-y_(1))hat(j)+(z_(2)-z_(2))hat(k)`
`=(4-1)hat(i)+(5-2)hat(j)+(6-3)hat(k)=3hat(i)+3hat(j)+3hat(k)`
Comparing with `X=xhat(i)+yhat(j)+zhat(k)`, we get `X=3,y=3,z=3`
`therefore` Magnitude of given vector `|PQ|=sqrt(3^2+3^2+3^2)=sqrt(9+9+9)=sqrt(27)=3sqrt(3)`
Hence, the unit vector in the direction of PQ.
`(vec(PQ))/(|vec(PQ)|)=(3hat(i)+3hatj+3hatk)/(3sqrt(3))=(3)/(3sqrt(3))(hat(i)+hat(j)+hat(k))=(1)/(sqrt(3))hat(i)+(1)/(sqrt(3))hat(j)+(1)/(sqrt(3))hat(k)`.