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.)
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!
To find two unit vectors that make an angle of 60° with vector v = 3, 4, we need to follow these steps:
1. Calculate the magnitude of vector v using the Pythagorean theorem:
magnitude(v) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
2. Divide vector v by its magnitude to obtain the corresponding unit vector:
unit vector(u) = v / magnitude(v)
= 3/5, 4/5
= 0.6, 0.8
3. Rotate the unit vector by the desired angle using the rotation matrix:
rotation matrix(R) = [cos θ, -sin θ]
[sin θ, cos θ]
For a 60° angle, the rotation matrix becomes:
R = [cos 60°, -sin 60°]
[sin 60°, cos 60°]
= [0.5, -√3/2]
[√3/2, 0.5]
4. Multiply the rotation matrix by the unit vector to obtain the two unit vectors that make an angle of 60° with v:
u1 = R * u
= [0.5, -√3/2] * [0.6, 0.8]
= [0.3 - √3/2, -0.3√3 + 0.4]
≈ [0.3 - 0.866, -0.519 + 0.4]
≈ [-0.566, -0.119]
u2 = R * u
= [0.5, -√3/2] * [0.6, 0.8]
= [0.3 - √3/2, -0.3√3 + 0.4]
≈ [0.3 - 0.866, -0.519 + 0.4]
≈ [-0.566, -0.119]
Therefore, the two unit vectors that make an angle of 60° with v are approximately:
-0.566, -0.119 and -0.566, -0.119