Dot Product

Let the required unit vector be n = cos@i + sin@j.

By dot product,

v‧n = │v││n│cos60*

As n is a unit vector, n = 1

3cos@ + 4sin@ = sqrt[(3^2) + (4^2)] (1) (1/2)

5sin(@ + sin^(-1)(3/5)) = 5/2

sin(@ + sin^(-1)(3/5)) = 1/2

@ + sin^(-1)(3/5) = 30* or 150*

@ = 30* - sin^(-1)(3/5) or 150* - sin^(-1)(3/5)

cos@ = cos30*cos[sin^(-1)(3/5)] + sin30*sin[sin^(-1)(3/5)]

= (sqrt3)/2 (4/5) + (1/2)(3/5)

= (4sqrt3 + 3)/10

or cos@ = cos150*cos[sin^(-1)(3/5)] + sin150*sin[sin^(-1)(3/5)]

= (-sqrt3)/2 (4/5) + (1/2)(3/5)

= (-4sqrt3 + 3)/10

sin@ = sin30*cos[sin^(-1)(3/5)] - cos30*sin[sin^(-1)(3/5)]

= (1/2)(4/5) - (sqrt3)/2 (3/5)

= (4 - 3sqrt3)/10

or sin@ = sin150*cos[sin^(-1)(3/5)] - cos150*sin[sin^(-1)(3/5)]

= (1/2)(4/5) + (sqrt3)/2 (3/5)

= (4 + 3sqrt3)/10

So, the unit vectors are:

(4sqrt3 + 3)/10 i + (4 - 3sqrt3)/10 j

or

(-4sqrt3 + 3)/10 i + (4 + 3sqrt3)/10 j