Find two unit vectors in 2-space that make an angle of 45◦ with 4i+3j.

First, lets transform the given vector into an unit vector (dividing by its module)

UnitVec = 4/5 i + 3/5 j

Then lets change this vector into a polar form

UnitVec = 1. with angle of 36.869 degrees taking as a reference the i vector

Then, the probem tells us that the vectors u and v make an angle of 45 degrees with UnitVec, so lets add+-45 to the vector in polar form

U = 1*[cos (36.869 + 45) i + sin (36.869 + 45) j] = 0.1414 i + 0.9899 j

V = 1*[cos (36.869 - 45) i + sin (36.869 - 45) j] = 0.9899 i - 0.1414 j