Find the angle between the vectors `hat(i)-2hat(j)+3hat(k) and 3hat(i)-2hat(j)+hat(k)`.

Explanation

Let `vec(a)=hat(i)-2hat(j)+3hat(k) and vec(b)=3hat(i)-2hat(j)+hat(k)`.
Magnitude of `vec(a),|vec(a)|=sqrt(1^2+(-2)^2+3^2)=sqrt(1+4+9)=sqrt(14)`.
Magnitude of `vec(b),|vec(b)|=sqrt(3^2+(-2)^2+1^2)=sqrt(9+4+1)=sqrt(14)`.
Now, `vec(a)vec(b)=(hat(i)-2hat(j)+3hat(k)).(3hat(i)-2hat(j)+hat(k))=1.3+(-2).(-2)+3.1=3+4+3=10`
(Do product of two vectors is equal to the sum of the products of their corresponding components.)
Let `theta` be the required angle between a and b then
`cos theta =(vec(a).vec(b))/(|vec(a)||vec(b)|)=(10)/(sqrt(14)sqrt(14))=(10)/(14)=(5)/(7)rArr theta= cos^(-1)((5)/(7))`.