Question

Find two unit vectors that make an angle of 60° with
v =

3, 4


(Enter your answer as a comma-separated list of vectors. Round your answers to four decimal places.)

Answer 1

Let's pick an arbitrary value for the first component, say 1.
then let the vector be (1,b)

(1,b)∙(3,4) = |(1,b)||(3,4)cos60°
3 + 4b = √(1+b^2)(5)(1/2)
6 + 8b = 5√(1+b^2)
36 + 96b + 64b^2 = 25(1+b^2) after squaring both sides
39b^2 + 96b + 11 = 0
Using the quadratic equation, I got
b = -.1205 or b = -2.3411

so one vector is (1,-.1205 , the other is (1, -2.3411)
however, when I sketched it, I noticed the second vector would make an angle of 120° , which is the supplement of 60°, so let's us
(-1, 2.3411)

also you needed unit vectors:
|(1, -.1205)} = 1.00723 , so we have (1/1.00723)times vector(1.-.1205)
so one vector is (.9928, - .1196)
in the same way, find the unit vector for (-1,2.3411)

I will check my first vector:
LS = (3,4)∙(.9928, -.1196) = 2.5
RS = |(.9928, -.1196)| |(3,4)cos60°
= 1(5)(1/2) = 2.5

My first vector works!