Let #vecv = 2hati+2hatj+2hatk#
The unit vector of any vector #vecv# is denoted by placing a hat, #hat#, (or caret) on top the letter #hatv#. The unit vector is obtained by dividing the vector by its magnitude:
#hatv = vecv/|vecv|" [1]"#
The magnitude of a vector is computed by taking the square root of the sum of the squares of its components:
#|vecv|= sqrt(v_x^2+v_y^2+v_z^2)" [2]"#
Substitute the components of vector, #vecv#, into equation [2]:
#|vecv|= sqrt(2^2+2^2+2^2)#
#|vecv|= sqrt12#
#|vecv|= 2sqrt3#
Substitute #vecv# and its magnitude into equation [1]:
#hatv = (2hati+2hatj+2hatk)/(2sqrt3)#
Divide each component:
#hatv = 1/sqrt3hati+1/sqrt3hatj+1/sqrt3hatk#
Rationalize the denominators:
#hatv = sqrt3/3hati+sqrt3/3hatj+sqrt3/3hatk#
To Check that this is a unit vector, we verify that its magnitude is 1:
#|hatv| = sqrt((sqrt3/3)^2+(sqrt3/3)^2+(sqrt3/3)^2)#
#|hatv| = sqrt(3/9+3/9+3/9)#
#|hatv| = sqrt(9/9)#
#|hatv| = sqrt(1) = 1#
This checks.