calculus
posted by colt on .
Find two unit vectors that make an angle of 60° with
v =
3, 4
(Enter your answer as a commaseparated list of vectors. Round your answers to four decimal places.)

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!